5.11. Konvergente Potenzreihen


In diesem Abschnitt verallgemeinern wir den Polynombegriff. Da ein Polynom die endliche Summe seiner Monome ist, liegt es nahe, unendliche Summen dieser Art, d.h. Funktionenreihen zu betrachten. Dies ist allerdings nur ein formaler Gedanke. Ob dadurch wieder eine Funktion, in die man reellen Zahlen einsetzen kann, gegeben ist, wird nur über Konvergenzbetrachtungen zu entscheiden sein.

Bei Polynomen lassen sich die Funktionswerte allein durch Addition und Multiplikation ermitteln; ein nicht zu unterschätzender Vorteil. Bei ihren Verallgemeinerungen wird dies so elementar nicht mehr möglich sein. Ist man jedoch nur an Näherungswerten interessiert, so darf man sich wieder auf polynomiale Verhältnisse zurück ziehen.

Definition:  Für jede Folge ( a n ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGPaWaaSbaaSqaaiaad6gacqGHLjYScaaIWaaabeaaaaa@3C70@ in MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@36D9@ und jedes a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgIGiolabl2riHcaa@3943@ nennen wir die Funktionenreihe

( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@
[5.11.1]

eine Potenzreihe mit Entwicklungspunkt  a.

Ist ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ eine Potenzreihe, so erhält man für jedes x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHcaa@395A@ durch Einsetzen die Zahlenreihe

( i=0 n a i (xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG4bGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43F2@

Für den Entwicklungspunkt a ist diese Reihe immer konvergent:  ( i=0 n a i (aa) i )=( a 0 ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGHbGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaGaeyypa0JaaiikaiaadggadaWgaaWcbaGaaGimaaqabaGccaGGPaWaaSbaaSqaaiaad6gacqGHLjYScaaIWaaabeaaaaa@4BAF@ . Interessant sind nun solche Potenzreihen, die in mindestens einem weiteren Punkt konvergieren.

Definition:  Wir nennen eine Potenzreihe ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@   konvergent, wenn sie in mindestens zwei Punkten konvergiert, wenn es also ein  ya MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabgcMi5kaadggaaaa@3914@ gibt, so dass

( i=0 n a i (ya) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG5bGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43F3@   konvergent ist.
[5.11.2]

Potenzreihen, die nur in ihrem Entwicklungspunkt konvergieren, heißen divergent.

Überraschenderweise zieht die Konvergenz in nur einem weiteren Punkt die Konvergenz in einer ganzen Umgebung von a nach sich.

Bemerkung:  Konvergiert die Potenzreihe ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ in einem  ya MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabgcMi5kaadggaaaa@3914@ , so konvergiert sie (sogar absolut) in jedem

x]a|ya|,a+|ya|[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaac2facaWGHbGaeyOeI0IaaiiFaiaadMhacqGHsislcaWGHbGaaiiFaiaacYcacaWGHbGaey4kaSIaaiiFaiaadMhacqGHsislcaWGHbGaaiiFaiaacUfaaaa@4797@
[5.11.3]

Beweis:  Da die Reihe ( i=0 n a i (ya) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG5bGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43F3@ konvergiert, ist nach [5.9.9] die Folge ( a n (ya) n ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamyEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaaiykamaaBaaaleaacaWGUbGaeyyzImRaaGimaaqabaaaaa@41C4@ eine Nullfolge, also insbesondere beschränkt. Es gibt also ein c, so dass

| a i (ya) i |c| a i | c |ya | i     für alle  i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamyEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaOGaaiiFaiabgsMiJkaadogacaaMf8Uaeyi1HSTaaGzbVlaacYhacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiiFaiabgsMiJoaalaaabaGaam4yaaqaaiaacYhacaWG5bGaeyOeI0IaamyyaiaacYhadaahaaWcbeqaaiaadMgaaaaaaOGaaeOzaiaabYpacaqGYbGaaeiiaiaabggacaqGSbGaaeiBaiaabwgacaWGPbGaeyicI4SaeSyfHukaaa@5F0C@

Dies erlaubt für alle i die Abschätzung

| a i (xa) i |=| a i ||xa | i c| xa ya | i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamiEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaOGaaiiFaiabg2da9iaacYhacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiiFaiaacYhacaWG4bGaeyOeI0IaamyyaiaacYhadaahaaWcbeqaaiaadMgaaaGccqGHKjYOcaWGJbGaeyyXICTaaiiFamaalaaabaGaamiEaiabgkHiTiaadggaaeaacaWG5bGaeyOeI0IaamyyaaaacaGG8bWaaWbaaSqabeaacaWGPbaaaaaa@5779@ .

Aus |xa|<|ya| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgYda8iaacYhacaWG5bGaeyOeI0IaamyyaiaacYhaaaa@400E@ folgt | xa ya |<1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFamaalaaabaGaamiEaiabgkHiTiaadggaaeaacaWG5bGaeyOeI0IaamyyaaaacaGG8bGaeyipaWJaaGymaaaa@3ED9@ , so dass die geometrische Reihe ( i=0 n c| xa ya | i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaam4yaiabgwSixlaacYhadaWcaaqaaiaadIhacqGHsislcaWGHbaabaGaamyEaiabgkHiTiaadggaaaGaaiiFamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@48A2@ eine konvergente Majorante zu ( i=0 n a i (xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG4bGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43F2@ ist, deren Konvergenz daher nach dem Majorantenkriterium [5.9.15] gesichert ist.

Konvergente Potenzreihen konvergieren also niemals nur in isolierten Punkten, sondern mindestens in einem offenen Intervall. Die folgende Definition liefert die Einzelheiten.

Definition:  Ist ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ eine Potenzreihe, so nennen wir die Zahl

rsup{|ya|||y      ( i=0 n a i (ya) i )  ist konvergent } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2da9iGacohacaGG1bGaaiiCaiaacUhacaGG8bGaamyEaiabgkHiTiaadggacaGG8bGaaiiFaiaadMhacqGHiiIZcqWIDesOcaaMe8Uaey4jIKTaaGjbVlaacIcadaaeWbqaaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamyEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aOGaaiykaiaabMgacaqGZbGaaeiDaiaabccacaqGRbGaae4Baiaab6gacaqG2bGaaeyzaiaabkhacaqGNbGaaeyzaiaab6gacaqG0bGaaiyFaaaa@663E@
[5.11.4]

ihren Konvergenzradius (dabei lassen wir auch den Wert r= MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2da9iabg6HiLcaa@38D7@  *) zu); für r>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg6da+iaaicdaaaa@3822@ nennen wir das offene Intervall

]ar,a+r[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiyxaiaadggacqGHsislcaWGYbGaaiilaiaadggacqGHRaWkcaWGYbGaai4waaaa@3D62@
[5.11.5]

ihren Konvergenzbereich.

___________
*) Wir setzen:  >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOhIuQaeyOpa4JaaGimaaaa@389C@   und für jedes reelle c ±c= MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOhIuQaeyySaeRaam4yaiabg2da9iabg6HiLcaa@3C27@ .

Beachte:

  • r=0( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2da9iaaicdacaaMf8Uaeyi1HSTaaGzbVlaacIcadaaeWbqaaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamiwaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aOGaaiykaaaa@4C01@   ist divergent.

  • r=( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2da9iaaicdacaaMf8Uaeyi1HSTaaGzbVlaacIcadaaeWbqaaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamiwaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aOGaaiykaaaa@4C01@   hat ganz MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@36D9@ als Konvergenzbereich.

  • |xa|>r( i=0 n a i (xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabg6da+iaadkhacaaMf8UaeyO0H4TaaGzbVlaacIcadaaeWbqaaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamiEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aOGaaiykaaaa@503A@   ist divergent.
     

Die in [5.11.4] gegebene Definition ist für die praktische Berechnung des Konvergenzradius r oft ungünstig. Andere Charakterisierungen von r sind daher von Vorteil.

Bemerkung:  Es sei ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ eine Potenzreihe mit Konvergenzradius r. Sind alle Koeffizienten a i 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbaabeaakiabgcMi5kaaicdaaaa@39F4@ , so gilt:

  1. | a n+1 a n |0r= MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFamaalaaabaGaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaaakeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaaakiaacYhacqGHsgIRcaaIWaGaaGzbVlabgkDiElaaywW7caWGYbGaeyypa0JaeyOhIukaaa@48C2@
[5.11.6]
  1. (| a n a n+1 | ) n0   konvergentr=lim| a n a n+1 | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacYhadaWcaaqaaiaadggadaWgaaWcbaGaamOBaaqabaaakeaacaWGHbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaaaaGccaGG8bGaaiykamaaBaaaleaacaWGUbGaeyyzImRaaGimaaqabaGccaqGRbGaae4Baiaab6gacaqG2bGaaeyzaiaabkhacaqGNbGaaeyzaiaab6gacaqG0bGaaGzbVlabgkDiElaaywW7caWGYbGaeyypa0JaciiBaiaacMgacaGGTbGaaiiFamaalaaabaGaamyyamaaBaaaleaacaWGUbaabeaaaOqaaiaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaaakiaacYhaaaa@5DA8@
[5.11.7]

Beweis:  Wir setzen mehrfach das Quotientenkriterium [5.9.16] ein und berechnen daher für xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgcMi5kaadggaaaa@3913@ :

| a i+1 (xa) i+1 a i (xa) i |=|xa|| a i+1 a i | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFamaalaaabaGaamyyamaaBaaaleaacaWGPbGaey4kaSIaaGymaaqabaGccaGGOaGaamiEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbGaey4kaSIaaGymaaaaaOqaaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamiEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaaaakiaacYhacqGH9aqpcaGG8bGaamiEaiabgkHiTiaadggacaGG8bGaeyyXICTaaiiFamaalaaabaGaamyyamaaBaaaleaacaWGPbGaey4kaSIaaGymaaqabaaakeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaaaakiaacYhaaaa@5944@
[0]
1.  

Mit (| a n+1 a n | ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacYhadaWcaaqaaiaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaaaaGccaGG8bGaaiykamaaBaaaleaacaWGUbGaeyyzImRaaGimaaqabaaaaa@422C@ ist für jedes x auch (|xa|| a n+1 a n | ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacYhacaWG4bGaeyOeI0IaamyyaiaacYhacqGHflY1caGG8bWaaSaaaeaacaWGHbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaaaOqaaiaadggadaWgaaWcbaGaamOBaaqabaaaaOGaaiiFaiaacMcadaWgaaWcbaGaamOBaiabgwMiZkaaicdaaeqaaaaa@4946@ eine Nullfolge. Also gibt es ein k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiolablwriLcaa@3949@ , so dass |xa|| a i+1 a i |< 1 2   für alle  ik MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgwSixlaacYhadaWcaaqaaiaadggadaWgaaWcbaGaamyAaiabgUcaRiaaigdaaeqaaaGcbaGaamyyamaaBaaaleaacaWGPbaabeaaaaGccaGG8bGaeyipaWZaaSaaaeaacaaIXaaabaGaaGOmaaaacaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaadMgacqGHLjYScaWGRbaaaa@521D@ . Nach Quotientenkriterium konvergiert die Potenzreihe daher in jedem xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgcMi5kaadggaaaa@3913@ .

2.  

Wir setzen zur Abkürzung  r lim| a n a n+1 |0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaafaGaeyypa0JaciiBaiaacMgacaGGTbGaaiiFamaalaaabaGaamyyamaaBaaaleaacaWGUbaabeaaaOqaaiaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaaakiaacYhacqGHLjYScaaIWaaaaa@448D@   und zeigen nun:

  • r r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaafaGaeyyzImRaamOCaaaa@3929@ :  Ist nämlich |xa|> r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabg6da+iqadkhagaqbaaaa@3C44@ , so gibt es ein k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiolablwriLcaa@3949@ , so dass (beachte: xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgcMi5kaadggaaaa@3913@ )

    | (xa) i+1 (xa) i |=|xa|>| a i a i+1 | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFamaalaaabaGaaiikaiaadIhacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaiabgUcaRiaaigdaaaaakeaacaGGOaGaamiEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaaaakiaacYhacqGH9aqpcaGG8bGaamiEaiabgkHiTiaadggacaGG8bGaeyOpa4JaaiiFamaalaaabaGaamyyamaaBaaaleaacaWGPbaabeaaaOqaaiaadggadaWgaaWcbaGaamyAaiabgUcaRiaaigdaaeqaaaaakiaacYhaaaa@5251@

    für alle ik MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgwMiZkaadUgaaaa@390D@ . Für diese i ist daher | a i+1 (xa) i+1 || a i (xa) i | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamyAaiabgUcaRiaaigdaaeqaaOGaaiikaiaadIhacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaiabgUcaRiaaigdaaaGccaGG8bGaeyyzImRaaiiFaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamiEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaOGaaiiFaaaa@4D19@ und (mit einem Induktionsargument)

    | a i (xa) i || a k (xa) k |>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamiEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaOGaaiiFaiabgwMiZkaacYhacaWGHbWaaSbaaSqaaiaadUgaaeqaaOGaaiikaiaadIhacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaam4AaaaakiaacYhacqGH+aGpcaaIWaaaaa@4BA5@ .

    ( a n (xa) n ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaaiykamaaBaaaleaacaWGUbGaeyyzImRaaGimaaqabaaaaa@41C3@ ist also keine Nullfolge, ( i=0 n a i (xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG4bGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43F2@ daher nicht konvergent. Wäre nun r <r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaafaGaeyipaWJaamOCaaaa@3867@ , so müsste es ein x mit |xa|> r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabg6da+iqadkhagaqbaaaa@3C44@ geben, in dem die Potenzreihe konvergiert. Dies ist aber nach unserer Überlegung nicht möglich.

  • r r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaafaGaeyizImQaamOCaaaa@3918@ :  Ist |xa|< r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgYda8iqadkhagaqbaaaa@3C40@ , so wähle man zwei Zahlen c 1 , c 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBaaaleaacaaIXaaabeaakiaacYcacaWGJbWaaSbaaSqaaiaaikdaaeqaaaaa@39C2@ mit |xa|< c 1 < c 2 < r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgYda8iaadogadaWgaaWcbaGaaGymaaqabaGccqGH8aapcaWGJbWaaSbaaSqaaiaaikdaaeqaaOGaeyipaWJabmOCayaafaaaaa@41FB@ . Man findet ein k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiolablwriLcaa@3949@ , so dass

    c 2 <| a i a i+1 | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBaaaleaacaaIYaaabeaakiabgYda8iaacYhadaWcaaqaaiaadggadaWgaaWcbaGaamyAaaqabaaakeaacaWGHbWaaSbaaSqaaiaadMgacqGHRaWkcaaIXaaabeaaaaGccaGG8baaaa@4008@

    für alle ik MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgwMiZkaadUgaaaa@390D@ . Für diese i schätzen wir nun ab:

    |xa|| a i+1 a i |= |xa| | a i a i+1 | < c 1 c 2 <1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgwSixlaacYhadaWcaaqaaiaadggadaWgaaWcbaGaamyAaiabgUcaRiaaigdaaeqaaaGcbaGaamyyamaaBaaaleaacaWGPbaabeaaaaGccaGG8bGaeyypa0ZaaSaaaeaacaGG8bGaamiEaiabgkHiTiaadggacaGG8baabaGaaiiFamaalaaabaGaamyyamaaBaaaleaacaWGPbaabeaaaOqaaiaadggadaWgaaWcbaGaamyAaiabgUcaRiaaigdaaeqaaaaakiaacYhaaaGaeyipaWZaaSaaaeaacaWGJbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaam4yamaaBaaaleaacaaIYaaabeaaaaGccqGH8aapcaaIXaaaaa@5871@ .

    Mit [0] folgt daher die Konvergenz der Potenzreihe in xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgcMi5kaadggaaaa@3913@ aus dem Quotientenkriterium. Wäre jetzt r >r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaafaGaeyOpa4JaamOCaaaa@386B@ , so wähle man ein x mit r<|xa|< r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabgYda8iaacYhacaWG4bGaeyOeI0IaamyyaiaacYhacqGH8aapceWGYbGbauaaaaa@3E3B@ , also ein x außerhalb des Konvergenzbereichs in dem die Potenzreihe konvergiert.

Beachte:

  • Da die Konvergenz einer Reihe nicht von ihren ersten Gliedern abhängt, gilt dieses Kriterium auch dann, wenn die Bedingung a i 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbaabeaakiabgcMi5kaaicdaaaa@39F4@ nur für fast alle i erfüllt ist.

  • Eine weitere Berechnungsmöglichkeit für r, die Formel von Hadamard, ergibt sich aus dem Wurzelkriterium [5.9.17]. Wir zitieren sie ohne Beweis:

    Ist die Folge ( | a n | n ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaakeaabaGaaiiFaiaadggadaWgaaWcbaGaamOBaaqabaGccaGG8baaleaacaWGUbaaaOGaaiykamaaBaaaleaacaWGUbGaeyyzImRaaGimaaqabaaaaa@3F88@ beschränkt, so besitzt sie einen größten Häufungspunkt, den sog. limes superior (in Zeichen lim ¯ | a n | n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaciGGSbGaaiyAaiaac2gaaaWaaOqaaeaacaGG8bGaamyyamaaBaaaleaacaWGUbaabeaakiaacYhaaSqaaiaad6gaaaaaaa@3D67@ ). Für den Konvergenzradius gilt dann:

    r={ 1 lim ¯ | a n | n  ,   falls   lim ¯ | a n | n 0  ,   falls   lim ¯ | a n | n =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6063@
    [5.11.8]

    Ist ( | a n | n ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaakeaabaGaaiiFaiaadggadaWgaaWcbaGaamOBaaqabaGccaGG8baaleaacaWGUbaaaOGaaiykamaaBaaaleaacaWGUbGaeyyzImRaaGimaaqabaaaaa@3F88@ unbeschränkt, so ist r=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2da9iaaicdaaaa@3820@ .


 

Bemerkung und Bezeichnung:  Eine konvergente Potenzreihe ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ konvergiert (absolut) in jedem Punkt x ihres Konvergenzbereichs. Die von ihr erzeugte Funktion

f:]ar,a+r[,f(x) i=0 a i (xa) i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaGGDbGaamyyaiabgkHiTiaadkhacaGGSaGaamyyaiabgUcaRiaadkhacaGGBbGaeyOKH4QaeSyhHeQaaiilaiaaywW7caWGMbGaaiikaiaadIhacaGGPaGaeyypa0ZaaabCaeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIhacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaaa@5691@
[5.11.9]

nennen wir ihre Grenzfunktion und bezeichnen sie mit dem Symbol  i=0 a i (Xa) i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaaa@42ED@ .

Beweis:  Ist x]ar,a+r[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaac2facaWGHbGaeyOeI0IaamOCaiaacYcacaWGHbGaey4kaSIaamOCaiaacUfaaaa@3FE3@ , also |xa|<r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgYda8iaadkhaaaa@3C34@ , so muss |xa| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaaaa@3A39@ von einem Element der Menge übertroffen werden, deren Supremum r ist. Es gibt also ein  y in dem ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ konvergiert, so dass

|xa|<|ya| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgYda8iaacYhacaWG5bGaeyOeI0IaamyyaiaacYhaaaa@400E@ .

Nach [5.11.3] ist daher ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ in x absolut konvergent.

Beachte:

  • Die Verhältnisse an den Rändern des Konvergenzbereichs sind nicht einheitlich zu beschreiben. Für die Potenzreihe ( i=0 n 1 i+1 X i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaSaaaeaacaaIXaaabaGaamyAaiabgUcaRiaaigdaaaGaamiwamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@41F2@ (mit Entwicklungspunkt 0) etwa errechnet man nach [5.11.7] den Konvergenzradius zu

    r=lim 1 n+1 1 n+2 =lim n+2 n+1 =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2da9iGacYgacaGGPbGaaiyBamaalaaabaWaaSaaaeaacaaIXaaabaGaamOBaiabgUcaRiaaigdaaaaabaWaaSaaaeaacaaIXaaabaGaamOBaiabgUcaRiaaikdaaaaaaiabg2da9iGacYgacaGGPbGaaiyBamaalaaabaGaamOBaiabgUcaRiaaikdaaeaacaWGUbGaey4kaSIaaGymaaaacqGH9aqpcaaIXaaaaa@4BC5@ .

    ( i=0 n 1 i+1 X i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaSaaaeaacaaIXaaabaGaamyAaiabgUcaRiaaigdaaaGaamiwamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@41F2@ konvergiert in 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaGymaaaa@3711@ (alternierende harmonische Reihe [5.9.12]), aber nicht in 1 (harmonische Reihe [5.9.6]).


     
  • Ist |x|<r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacaGG8bGaeyipaWJaamOCaaaa@3A61@ , so liegt x+a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgUcaRiaadggaaaa@382E@ im Konvergenzbereich ]ar,a+r[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiyxaiaadggacqGHsislcaWGYbGaaiilaiaadggacqGHRaWkcaWGYbGaai4waaaa@3D62@ . Daher ist die Reihe

    ( i=0 n a i x i )=( i=0 n a i (x+aa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaadIhadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaGaeyypa0JaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG4bGaey4kaSIaamyyaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aOGaaiykaaaa@521D@
    [5.11.10]

    absolut konvergent.


     

Beispiel:  

  • ( i=0 n (1) i (X1) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGPbaaaOGaaiikaiaadIfacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@45C3@ ist eine Potenzreihe mit Entwicklungspunkt 1. Nach [5.11.7] hat sie den Konvergenzradius 1 und damit den Konvergenzbereich ]0,2[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiyxaiaaicdacaGGSaGaaGOmaiaacUfaaaa@394F@ . Als Grenzfunktion erhält man über den Limes der geometrischen Reihe ([5.9.4]) die Einschränkung der Kehrwertfunktion auf ]0,2[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiyxaiaaicdacaGGSaGaaGOmaiaacUfaaaa@394F@ , denn für x]0,2[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaac2facaaIWaGaaiilaiaaikdacaGGBbaaaa@3BD0@ ist der Abstand |1x|<1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaaigdacqGHsislcaWG4bGaaiiFaiabgYda8iaaigdaaaa@3BCD@ , also hat man:

    i=0 (1) i (x1) i = i=0 (1x) i = 1 1(1x) = 1 x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaadMgaaaGccaGGOaGaamiEaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGPbaaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaeyOhIukaniabggHiLdGccqGH9aqpdaaeWbqaaiaacIcacaaIXaGaeyOeI0IaamiEaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoakiabg2da9maalaaabaGaaGymaaqaaiaaigdacqGHsislcaGGOaGaaGymaiabgkHiTiaadIhacaGGPaaaaiabg2da9maalaaabaGaaGymaaqaaiaadIhaaaaaaa@5BCC@
    [5.11.11]

     
  • Die Potenzreihen

    ( i=0 n 1 i! X i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaSaaaeaacaaIXaaabaGaamyAaiaacgcaaaGaamiwamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@40FA@ ,   ( i=0 n (1) i (2i+1)! X 2i+1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaSaaaeaacaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaadMgaaaaakeaacaGGOaGaaGOmaiaadMgacqGHRaWkcaaIXaGaaiykaiaacgcaaaGaamiwamaaCaaaleqabaGaaGOmaiaadMgacqGHRaWkcaaIXaaaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aOGaaiykaaaa@4A70@ ,   ( i=0 n (1) i (2i)! X 2i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaSaaaeaacaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaadMgaaaaakeaacaGGOaGaaGOmaiaadMgacaGGPaGaaiyiaaaacaWGybWaaWbaaSqabeaacaaIYaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@4736@
    [5.11.12]

    mit Entwicklungspunkt 0 konvergieren nach [5.9.18], [5.9.19], [5.9.20] auf ganz MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@36D9@ . Ihre Grenzfunktionen sind exp, sin und cos.


     
  • Ist p= i=0 k a i X i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2da9maaqahabaGaamyyamaaBaaaleaacaWGPbaabeaaieaakiaa=HfadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGRbaaniabggHiLdaaaa@4140@ irgendein Polynom, so wird durch die Festsetzung a i 0i>k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbaabeaakiabg2da9iaaicdacaqGSaGaamyAaiabg6da+iaadUgaaaa@3CC8@ , der Koeffizientensatz a 0 ,, a k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIWaaabeaakiaacYcacqWIMaYscaGGSaGaamyyamaaBaaaleaacaWGRbaabeaaaaa@3BC3@ zu einer Folge erweitert. Die Potenzreihe ( i=0 n a i X i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaaieaakiaa=HfadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@40AB@ konvergiert in ganz MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@36D9@ und die Grenzfunktion

    i=0 a i X i = i=0 k a i X i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaamiwamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aOGaeyypa0ZaaabCaeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaamiwamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaadUgaa0GaeyyeIuoaaaa@4AA8@
    [5.11.13]

    stimmt mit  p überein. Polymone sind also Grenzfunktionen spezieller Potenzreihen!

Wir befassen uns nun mit den Eigenschaften konvergenter Potenzreihen. Zunächst gelingt es, den Identitätssatz für Polynome (siehe [4.5.*]) in geeigneter Weise auf konvergente Potenzreihen zu übertragen: Eine Grenzfunktion ist bereits dann die Nullfunktion wenn sie abzählbar viele (geeignete) Nullstellen besitzt!

Bemerkung (Identitätssatz für konvergente Potenzreihen):   ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ sei eine konvergente Potenzreihe und ( x n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@38E8@ eine Folge in ]ar,a+r[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiyxaiaadggacqGHsislcaWGYbGaaiilaiaadggacqGHRaWkcaWGYbGaai4waaaa@3D62@ mit a x n a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgcMi5kaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsgIRcaWGHbaaaa@3D0F@ . Sind alle Folgenglieder Nullstellen der Grenzfunktion, so ist:

a k =0  für alle  k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGRbaabeaakiabg2da9iaaicdacaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaabccacaWGRbGaeyicI4SaeSyfHukaaa@4562@
[5.11.14]

Beweis:  Da x n a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkziUkaadggaaaa@3A62@ , findet man ein positives s<r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiabgYda8iaadkhaaaa@385C@ , so dass x n [as,a+s]  für alle  n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGUbaabeaakiabgIGiolaacUfacaWGHbGaeyOeI0Iaam4CaiaacYcacaWGHbGaey4kaSIaam4Caiaac2facaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaaaaa@4DB7@ . Nach [5.11.10] existiert der Grenzwert  c i=0 | a i | s i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabg2da9maaqahabaGaaiiFaiaadggadaWgaaWcbaGaamyAaaqabaGccaGG8bGaam4CamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaaa@43CA@ . Damit schätzen wir zunächst für alle x mit |xa|s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgsMiJkaadohaaaa@3CE6@ und alle  j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgIGiolablwriLcaa@3948@ folgendermaßen ab:

| i=j a i (xa) ij | i=j | a i | s ij = 1 s j i=j | a i | s i c s j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6CCD@
[1]

Wir beweisen nun [5.11.14] per Induktion:

  • Da alle Folgenglieder Nullstellen der Grenzfunktion sind, hat man mit  j=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabg2da9iaaigdaaaa@3819@ in [1] für alle n:

    0 = i=0 a i ( x n a) i = a 0 +( x n a) i=1 a i ( x n a) i1 | a 0 | =|( x n a) i=1 a i ( x n a) i1 | | x n a| c s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8A9D@

    Die Konvergenz  x n a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkziUkaadggaaaa@3A62@   erzwingt daher: a 0 =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIWaaabeaakiabg2da9iaaicdaaaa@38FF@ .

  • Sei nun bereits a 0 == a k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIWaaabeaakiabg2da9iablAciljabg2da9iaadggadaWgaaWcbaGaam4AaaqabaGccqGH9aqpcaaIWaaaaa@3E39@ . Wir benutzen [1] für  j=k+2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabg2da9iaadUgacqGHRaWkcaaIYaaaaa@39EC@ und erhalten damit für alle n:

    0 = i=k+1 a i ( x n a) i = ( x n a) k+1 0 i=k+1 a i ( x n a) ik1 0 = i=k+1 a i ( x n a) ik1 = a k+1 +( x n a) i=k+2 a i ( x n a) ik2 | a k+1 | =|( x n a) i=k+2 a i ( x n a) ik2 | | x n a| c s k+2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabyWaaaaabaaabaGaaGimaaqaaiabg2da9maaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG4bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaadUgacqGHRaWkcaaIXaaabaGaeyOhIukaniabggHiLdaakeaaaeaaaeaacqGH9aqpdaagaaqaaiaacIcacaWG4bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadUgacqGHRaWkcaaIXaaaaaqaaiabgcMi5kaaicdaaOGaayjo+dWaaabCaeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaiabgkHiTiaadUgacqGHsislcaaIXaaaaaqaaiaadMgacqGH9aqpcaWGRbGaey4kaSIaaGymaaqaaiabg6HiLcqdcqGHris5aaGcbaGaeyO0H4nabaGaaGimaaqaaiabg2da9maaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG4bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgacqGHsislcaWGRbGaeyOeI0IaaGymaaaaaeaacaWGPbGaeyypa0Jaam4AaiabgUcaRiaaigdaaeaacqGHEisPa0GaeyyeIuoaaOqaaaqaaaqaaiabg2da9iaadggadaWgaaWcbaGaam4AaiabgUcaRiaaigdaaeqaaOGaey4kaSIaaiikaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGHbGaaiykamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG4bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgacqGHsislcaWGRbGaeyOeI0IaaGOmaaaaaeaacaWGPbGaeyypa0Jaam4AaiabgUcaRiaaikdaaeaacqGHEisPa0GaeyyeIuoaaOqaaiabgkDiEdqaaiaacYhacaWGHbWaaSbaaSqaaiaadUgacqGHRaWkcaaIXaaabeaakiaacYhaaeaacqGH9aqpcaGG8bGaaiikaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGHbGaaiykamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG4bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgacqGHsislcaWGRbGaeyOeI0IaaGOmaaaaaeaacaWGPbGaeyypa0Jaam4AaiabgUcaRiaaikdaaeaacqGHEisPa0GaeyyeIuoakiaacYhaaeaaaeaaaeaacqGHKjYOcaGG8bGaamiEamaaBaaaleaacaWGUbaabeaakiabgkHiTiaadggacaGG8bGaeyyXIC9aaSaaaeaacaWGJbaabaGaam4CamaaCaaaleqabaGaam4AaiabgUcaRiaaikdaaaaaaaaaaaa@D43E@

    Die Gleichheit a k+1 =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaGccqGH9aqpcaaIWaaaaa@3AD2@ folgt nun nun wieder aus der Konvergenz  x n a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkziUkaadggaaaa@3A62@ .

Beachte:

  • [5.11.14] benutzt man oft beim sog. Koeffizientenvergleich:

    Findet man zu zwei konvergenten Potenzreihen ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ und ( i=0 n b i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamOyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D3@ eine Folge ( x n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@38E8@ im Schnitt der Konvergenzbereiche mit a x n a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgcMi5kaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsgIRcaWGHbaaaa@3D0F@ , so dass i=0 a i ( x n a) i = i=0 b i ( x n a) i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aOGaeyypa0ZaaabCaeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaaa@5414@ für alle n, so sind die beiden Potenzreihen bereits identisch, d.h.

    a k = b k   für alle  k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGRbaabeaakiabg2da9iaadkgadaWgaaWcbaGaam4AaaqabaGccaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaadUgacqGHiiIZcqWIvesPaaa@4612@
    [5.11.15]

    Beweis:  Die Potenzreihe ( i=0 n ( a i b i ) (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaaiikaiaadggadaWgaaWcbaGaamyAaaqabaGccqGHsislcaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiykaiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@4823@ (siehe [5.11.17]) erfüllt die Bedingung in [5.11.14].
     

Die nächste Bemerkung bestätigt die Verträglichkeit der Konvergenz von Potenzreihen mit den vier Grundrechenarten.

Bemerkung:   ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ und ( i=0 n b i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamOyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D3@ seien zwei konvergente Potenzreihen mit den Konvergenzradien r 1   und   r 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaaIXaaabeaakiaabwhacaqGUbGaaeizaiaadkhadaWgaaWcbaGaaGOmaaqabaaaaa@3C00@ . Dann sind auch ihre Summe, ihre Differenz und ihr Produkt konvergente Potenzreihen. Für |xa|<min{ r 1 , r 2 } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgYda8iGac2gacaGGPbGaaiOBaiaacUhacaWGYbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadkhadaWgaaWcbaGaaGOmaaqabaGccaGG9baaaa@4490@ gilt darüber hinaus:

  1. i=0 ( a i + b i ) (xa) i = i=0 a i (xa) i + i=0 b i (xa) i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaGGOaGaamyyamaaBaaaleaacaWGPbaabeaakiabgUcaRiaadkgadaWgaaWcbaGaamyAaaqabaGccaGGPaGaaiikaiaadIhacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aOGaeyypa0ZaaabCaeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIhacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aOGaey4kaSYaaabCaeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIhacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaaa@6498@
[5.11.16]
  1. i=0 ( a i b i ) (xa) i = i=0 a i (xa) i i=0 b i (xa) i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@64AE@
[5.11.17]
  1. i=0 ( j=0 i a j b ij ) (xa) i = i=0 a i (xa) i i=0 b i (xa) i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6CCF@
[5.11.18]
  1. Ist b 0 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIWaaabeaakiabgcMi5kaaicdaaaa@39C1@ , so ist auch ( i=0 n a i (Xa) i ) ( i=0 n b i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaGGOaWaaabCaeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaeaacaGGOaWaaabCaeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaaaaa@524C@ eine konvergente Potenzreihe.
[5.11.19]

Beweis:  Da Reihen spezielle Folgen sind, erhält man 1., 2. und 3. aus den ersten drei Grenzwertsätzen [5.6.1] - [5.6.3].

In 4. reicht es, den Quotienten 1 ( i=0 n b i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaiikamaaqahabaGaamOyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaaaa@449E@ als konvergente Potenzreihe darzustellen. Dazu konstruieren wir zunächst eine Potenzreihe ( i=0 n c i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaam4yamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D4@ , so dass

( i=0 n b i (Xa) i )( i=0 n c i (Xa) i )= (1) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamOyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaGaeyyXICTaaiikamaaqahabaGaam4yamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaGaeyypa0JaaiikaiaaigdacaGGPaWaaSbaaSqaaiaad6gacqGHLjYScaaIWaaabeaaaaa@5B41@
[2]

Dies gelingt, wenn man Koeffizienten c i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBaaaleaacaWGPbaabeaaaaa@376B@ finden kann, so dass  j=0 i b j c ij ={ 1 , falls  i=0 0 , falls  i>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGIbWaaSbaaSqaaiaadQgaaeqaaOGaam4yamaaBaaaleaacaWGPbGaeyOeI0IaamOAaaqabaaabaGaamOAaiabg2da9iaaicdaaeaacaWGPbaaniabggHiLdGccqGH9aqpdaGabaqaauaabaqaceaaaeaacaaIXaGaaeOzaiaabggacaqGSbGaaeiBaiaabohacaWGPbGaeyypa0JaaGimaaqaaiaaicdacaqGMbGaaeyyaiaabYgacaqGSbGaae4CaiaadMgacqGH+aGpcaaIWaaaaaGaay5Eaaaaaa@5372@ .
Wir setzen daher rekursiv

c 0 1 b 0 c i+1 1 b 0 j=1 i+1 b j c i+1j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBaaaleaacaaIWaaabeaakiabg2da9maalaaabaGaaGymaaqaaiaadkgadaWgaaWcbaGaaGimaaqabaaaaOGaaGzbVlabgEIizlaaywW7caWGJbWaaSbaaSqaaiaadMgacqGHRaWkcaaIXaaabeaakiabg2da9iabgkHiTmaalaaabaGaaGymaaqaaiaadkgadaWgaaWcbaGaaGimaaqabaaaaOWaaabCaeaacaWGIbWaaSbaaSqaaiaadQgaaeqaaOGaam4yamaaBaaaleaacaWGPbGaey4kaSIaaGymaiabgkHiTiaadQgaaeqaaaqaaiaadQgacqGH9aqpcaaIXaaabaGaamyAaiabgUcaRiaaigdaa0GaeyyeIuoaaaa@56EA@

und haben damit [2], denn:

  • j=0 0 b j c 0j = b 0 c 0 =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGIbWaaSbaaSqaaiaadQgaaeqaaOGaam4yamaaBaaaleaacaaIWaGaeyOeI0IaamOAaaqabaaabaGaamOAaiabg2da9iaaicdaaeaacaaIWaaaniabggHiLdGccqGH9aqpcaWGIbWaaSbaaSqaaiaaicdaaeqaaOGaam4yamaaBaaaleaacaaIWaaabeaakiabg2da9iaaigdaaaa@473F@
     
  • j=0 i+1 b j c i+1j = b 0 c i+1 + j=1 i+1 b j c i+1j =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGIbWaaSbaaSqaaiaadQgaaeqaaOGaam4yamaaBaaaleaacaWGPbGaey4kaSIaaGymaiabgkHiTiaadQgaaeqaaaqaaiaadQgacqGH9aqpcaaIWaaabaGaamyAaiabgUcaRiaaigdaa0GaeyyeIuoakiabg2da9iaadkgadaWgaaWcbaGaaGimaaqabaGccaWGJbWaaSbaaSqaaiaadMgacqGHRaWkcaaIXaaabeaakiabgUcaRmaaqahabaGaamOyamaaBaaaleaacaWGQbaabeaakiaadogadaWgaaWcbaGaamyAaiabgUcaRiaaigdacqGHsislcaWGQbaabeaaaeaacaWGQbGaeyypa0JaaGymaaqaaiaadMgacqGHRaWkcaaIXaaaniabggHiLdGccqGH9aqpcaaIWaaaaa@5C96@

Da b 0 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIWaaabeaakiabgcMi5kaaicdaaaa@39C1@ , hat die Grenzfunktion i=0 b i (Xa) i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaaa@42EE@ im Entwicklungspunkt a einen Wert 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiyIKRaaGimaaaa@37EA@ . Wir zeigen nun, dass dies auch in einer Umgebung ]a 1 n ,a+ 1 n [ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiyxaiaadggacqGHsisldaWcaaqaaiaaigdaaeaacaWGUbaaaiaacYcacaWGHbGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGBbaaaa@3EF0@ von a so bleibt: Angenommen, zu jedem natürlichen n> 1 r 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg6da+maalaaabaGaaGymaaqaaiaadkhadaWgaaWcbaGaaGOmaaqabaaaaaaa@3A0E@ gibt es ein x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGUbaabeaaaaa@3785@ mit | x n a|< 1 n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGHbGaaiiFaiabgYda8maalaaabaGaaGymaaqaaiaad6gaaaaaaa@3E24@ , so dass i=0 b i ( x n a) i =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aOGaeyypa0JaaGimaaaa@4601@ . Da a x n a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgcMi5kaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsgIRcaWGHbaaaa@3D0F@ , folgt aus [5.11.14] b k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGRbaabeaakiabg2da9iaaicdaaaa@3936@ für alle k im Widerspruch zu b 0 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIWaaabeaakiabgcMi5kaaicdaaaa@39C1@ .

Es gibt also eine Umgebung von a auf der die Grenzfunktion i=0 b i (Xa) i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaaa@42EE@ niemals den Wert 0 annimmt. Für alle x aus dieser Umgebung ist daher in [2] der 4. Grenzwertsatz anwendbar:

i=0 n c i (xa) i = 1 i=0 n b i (xa) i 1 i=0 b i (xa) i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@61FA@

( i=0 n c i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaam4yamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D4@ ist somit konvergent und stellt gemäß [2] den Kehrwert 1 ( i=0 n b i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaiikamaaqahabaGaamOyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaaaa@449E@ dar.

Potenzreihen haben per Definition einen fest vorgewählten Entwicklungspunkt. Interessanterweise darf man ihn durch jeden Punkt des Konvergenzbereichs (lokal) austauschen. Der folgende Satz ist für weitere Überlegungen von großer technischer Bedeutung.

Bemerkung (Umordnungssatz):   ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ sei eine konvergente Potenzreihe, r>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg6da+iaaicdaaaa@3822@ ihr Konvergenzradius. Dann gilt für alle b]ar,a+r[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabgIGiolaac2facaWGHbGaeyOeI0IaamOCaiaacYcacaWGHbGaey4kaSIaamOCaiaacUfaaaa@3FCD@

Es gibt eine konvergente Potenzreihe ( i=0 n b i (Xb) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamOyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamOyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D4@ , so dass

i=0 b i (xb) i = i=0 a i (xa) i    für alle x mit  |xb|<sr|ba| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6BB8@
[5.11.20]

Beweis:  Man beachte zunächst, dass s>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg6da+iaaicdaaaa@3823@ ist. Ferner hat man für alle |xb|<s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGIbGaaiiFaiabgYda8iaadohaaaa@3C36@ :

|xa||ba|+|xb|<|ba|+s=r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgsMiJkaacYhacaWGIbGaeyOeI0IaamyyaiaacYhacqGHRaWkcaGG8bGaamiEaiabgkHiTiaadkgacaGG8bGaeyipaWJaaiiFaiaadkgacqGHsislcaWGHbGaaiiFaiabgUcaRiaadohacqGH9aqpcaWGYbaaaa@4FF0@
[3]

Für ein solches x motiviert die Rechnung (wir benutzen dabei das allgemeine Binomialtheorem [5.2.5])

i=0 n a i (xa) i = i=0 n a i ((ba)+(xb)) i = i=0 n j=0 i a i (T i j )T (ba) ij (xb) j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7594@
[4]

die Betrachtung der Doppelreihe ( i=0 n j=0 m b ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaacaWGIbWaaSbaaSqaaiaadMgacaaMc8UaamOAaaqabaaabaGaamOAaiabg2da9iaaicdaaeaacaWGTbaaniabggHiLdaaleaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@4702@   mit  b ij { a i (T i j )T (ba) ij (xb) j   , falls  ij 0  , falls  i<j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGPbGaaGPaVlaadQgaaeqaaOGaeyypa0ZaaiqaaeaafaqaaeGabaaabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcafaqabeGabaaabaGaamyAaaqaaiaadQgaaaGaaiykaiaacIcacaWGIbGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgacqGHsislcaWGQbaaaOGaaiikaiaadIhacqGHsislcaWGIbGaaiykamaaCaaaleqabaGaamOAaaaakiaabAgacaqGHbGaaeiBaiaabYgacaqGZbGaamyAaiabgwMiZkaadQgaaeaacaaIWaGaaeOzaiaabggacaqGSbGaaeiBaiaabohacaWGPbGaeyipaWJaamOAaaaaaiaawUhaaaaa@5E4B@

Wegen [3] ist gemäß [5.11.10] die Reihe ( i=0 n | a i | (|ba|+|xb|) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaaiiFaiaadggadaWgaaWcbaGaamyAaaqabaGccaGG8bGaaiikaiaacYhacaWGIbGaeyOeI0IaamyyaiaacYhacqGHRaWkcaGG8bGaamiEaiabgkHiTiaadkgacaGG8bGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@4D8F@ konvergent. Die für alle n,m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGTbGaeyicI4SaeSyfHukaaa@3AEE@ gültige Abschätzung

i=0 n j=0 m | b ij | i=0 n j=0 i | b ij | = i=0 n | a i | (|ba|+|xb|) i i=0 | a i | (|ba|+|xb|) i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8C9F@

belegt die Beschränktheit der Doppelreihe ( i=0 n j=0 m | b ij | ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaacaGG8bGaamOyamaaBaaaleaacaWGPbGaaGPaVlaadQgaaeqaaOGaaiiFaaWcbaGaamOAaiabg2da9iaaicdaaeaacaWGTbaaniabggHiLdaaleaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@4917@ . Nach [5.10.26] darf man daher bei der Limesberechnung der Doppelreihe ( i=0 n j=0 m b ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaacaWGIbWaaSbaaSqaaiaadMgacaaMc8UaamOAaaqabaaabaGaamOAaiabg2da9iaaicdaaeaacaWGTbaaniabggHiLdaaleaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@4702@ die Reihenfolge von Zeilen- und Spaltengrenzwert vertauschen, man hat also (beachte [3] und [4]):

i=0 a i (xa) i = i=0 j=0 i b ij = i=0 j=0 b ij = j=0 i=0 b ij = j=0 i=j a i (T i j )T (ba) ij b j (xb) j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@9E00@

5.10. 5.12.