5.11. Convergent Power Series


This chapter is up to a certain generalization of the polynomial idea. As a polynomial is the finit sum of its monoms, it is advisable to consider infinite sums of this kind, i.e. we are going to study series of functions. Of course this is just a formal concept at this stage. Whether it results in a genuine function, capable to accept real numbers, surely depends on convergence properties.

It is an advantage that polynomials calculate only with addition and multiplication. Although this simplicity certainly will get lost with our prospective generalized polynomials the original advantage will apply again if we are only interested in approximative values.

Definition:  For every sequence ( a n ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGPaWaaSbaaSqaaiaad6gacqGHLjYScaaIWaaabeaaaaa@3C70@ in MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@36D9@ and every a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgIGiolabl2riHcaa@3943@ the series of functions

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

is called a power series with  a as expansion point.

For every x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHcaa@395A@ the power series ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ provides a series of  numbers by evaluating X at x:

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

This series is always convergent for the expansion point a ( i=0 n a i (aa) i )=( a 0 ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGHbGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaGaeyypa0JaaiikaiaadggadaWgaaWcbaGaaGimaaqabaGccaGGPaWaaSbaaSqaaiaad6gacqGHLjYScaaIWaaabeaaaaa@4BAF@ . Thus we will focus on those power series which converge for at least one further number.

Definition:  We call a power series ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@   convergent, if it converges for at least two different points, i.e. if there is an  ya MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabgcMi5kaadggaaaa@3914@ such that

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

Power series which are convergent for their expansion point solely are called divergent.

Surprisingly the convergence for only one additional point leads to the convergence for a whole neighbourhood of a.

Proposition:  If ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ converges in a  ya MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabgcMi5kaadggaaaa@3914@ it converges (even absolutly) for every

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

Proof:  As the series ( i=0 n a i (ya) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG5bGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43F3@ converges, the sequence ( a n (ya) n ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamyEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaaiykamaaBaaaleaacaWGUbGaeyyzImRaaGimaaqabaaaaa@41C4@ is a zero sequence according to [5.9.9], thus bounded. So there is a c such that

| a i (ya) i |c| a i | c |ya | i     for all  i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamyEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaOGaaiiFaiabgsMiJkaadogacaaMf8Uaeyi1HSTaaGzbVlaacYhacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiiFaiabgsMiJoaalaaabaGaam4yaaqaaiaacYhacaWG5bGaeyOeI0IaamyyaiaacYhadaahaaWcbeqaaiaadMgaaaaaaOGaaeOzaiaabYpacaqGYbGaaeiiaiaabggacaqGSbGaaeiBaiaabwgacaWGPbGaeyicI4SaeSyfHukaaa@5F0C@ .

Thus the following estimate holds for all i:

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

From |xa|<|ya| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgYda8iaacYhacaWG5bGaeyOeI0IaamyyaiaacYhaaaa@400E@ we have | xa ya |<1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFamaalaaabaGaamiEaiabgkHiTiaadggaaeaacaWG5bGaeyOeI0IaamyyaaaacaGG8bGaeyipaWJaaGymaaaa@3ED9@ . Consequently the geometric series ( i=0 n c| xa ya | i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaam4yaiabgwSixlaacYhadaWcaaqaaiaadIhacqGHsislcaWGHbaabaGaamyEaiabgkHiTiaadggaaaGaaiiFamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@48A2@ converges and dominates the series ( i=0 n a i (xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG4bGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43F2@ which is convergent now due to the comparison test [5.9.15].

So we know that a convergent power series never converges for isolated points only. In fact its domain of convergence will always be an open interval, which allows to introduce the following notions.

Definition:  For any power series ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ the number

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

is called its radius of convergence (we liberally allow r= MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2da9iabg6HiLcaa@38D7@  *) as a value). If r>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg6da+iaaicdaaaa@3822@ we call the open interval

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

the domain of convergence.

___________
*) We set:  >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOhIuQaeyOpa4JaaGimaaaa@389C@   and  ±c= MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOhIuQaeyySaeRaam4yaiabg2da9iabg6HiLcaa@3C27@ for every real number c.

Consider:

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

  • r=( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2da9iaaicdacaaMf8Uaeyi1HSTaaGzbVlaacIcadaaeWbqaaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamiwaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aOGaaiykaaaa@4C01@   the domain of convergence is the whole of MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@36D9@ .

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

Calculating a radius of convergence according to [5.11.4] is an akward task, so there is a need for other techniques.

Proposition:  Let ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ be a power series and r its radius of convergence. If a i 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbaabeaakiabgcMi5kaaicdaaaa@39F4@ for all the coefficients, we have:

  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   convergentr=lim| a n a n+1 | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacYhadaWcaaqaaiaadggadaWgaaWcbaGaamOBaaqabaaakeaacaWGHbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaaaaGccaGG8bGaaiykamaaBaaaleaacaWGUbGaeyyzImRaaGimaaqabaGccaqGRbGaae4Baiaab6gacaqG2bGaaeyzaiaabkhacaqGNbGaaeyzaiaab6gacaqG0bGaaGzbVlabgkDiElaaywW7caWGYbGaeyypa0JaciiBaiaacMgacaGGTbGaaiiFamaalaaabaGaamyyamaaBaaaleaacaWGUbaabeaaaOqaaiaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaaakiaacYhaaaa@5DA8@
[5.11.7]

Proof:  We employ the ratio test [5.9.16] several times. Note that if xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgcMi5kaadggaaaa@3913@ we may calculate

| 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.  

From a n+1 a n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaaaOqaaiaadggadaWgaaWcbaGaamOBaaqabaaaaOGaeyOKH4QaaGimaaaa@3DDB@ we see that for each x the sequence (|xa|| a n+1 a n | ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacYhacaWG4bGaeyOeI0IaamyyaiaacYhacqGHflY1caGG8bWaaSaaaeaacaWGHbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaaaOqaaiaadggadaWgaaWcbaGaamOBaaqabaaaaOGaaiiFaiaacMcadaWgaaWcbaGaamOBaiabgwMiZkaaicdaaeqaaaaa@4946@ is a zero sequence as well. So there is a number k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiolablwriLcaa@3949@ such that |xa|| a i+1 a i |< 1 2   for all  ik MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgwSixlaacYhadaWcaaqaaiaadggadaWgaaWcbaGaamyAaiabgUcaRiaaigdaaeqaaaGcbaGaamyyamaaBaaaleaacaWGPbaabeaaaaGccaGG8bGaeyipaWZaaSaaaeaacaaIXaaabaGaaGOmaaaacaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaadMgacqGHLjYScaWGRbaaaa@521D@ . Due to the ratio test the power series thus converges for every xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgcMi5kaadggaaaa@3913@ .

2.  

We set as an abbreviation  r lim| a n a n+1 |0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaafaGaeyypa0JaciiBaiaacMgacaGGTbGaaiiFamaalaaabaGaamyyamaaBaaaleaacaWGUbaabeaaaOqaaiaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaaakiaacYhacqGHLjYScaaIWaaaaa@448D@   and prove:

  • r r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaafaGaeyyzImRaamOCaaaa@3929@ :  If |xa|> r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabg6da+iqadkhagaqbaaaa@3C44@ we find a number k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiolablwriLcaa@3949@ such that (note: 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@

    for all ik MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgwMiZkaadUgaaaa@390D@ . Hence we have for these i | a i+1 (xa) i+1 || a i (xa) i | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamyAaiabgUcaRiaaigdaaeqaaOGaaiikaiaadIhacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaiabgUcaRiaaigdaaaGccaGG8bGaeyyzImRaaiiFaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamiEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGPbaaaOGaaiiFaaaa@4D19@ and thus (by an easy induction argument)

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

    So ( a n (xa) n ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiEaiabgkHiTiaadggacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaaiykamaaBaaaleaacaWGUbGaeyyzImRaaGimaaqabaaaaa@41C3@ is no zero sequence and consequently the series ( i=0 n a i (xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG4bGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43F2@ is divergent. Now if r <r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaafaGaeyipaWJaamOCaaaa@3867@ , we would find an x with |xa|> r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabg6da+iqadkhagaqbaaaa@3C44@ such that the power series is convergent for, which contradicts the result above.

  • r r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaafaGaeyizImQaamOCaaaa@3918@ :  If |xa|< r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgYda8iqadkhagaqbaaaa@3C40@ we choose two numbers c 1 , c 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBaaaleaacaaIXaaabeaakiaacYcacaWGJbWaaSbaaSqaaiaaikdaaeqaaaaa@39C2@ such that |xa|< c 1 < c 2 < r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgYda8iaadogadaWgaaWcbaGaaGymaaqabaGccqGH8aapcaWGJbWaaSbaaSqaaiaaikdaaeqaaOGaeyipaWJabmOCayaafaaaaa@41FB@ . From the limit properties of  r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaafaaaaa@366C@ we get a  k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiolablwriLcaa@3949@ such that

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

    for all ik MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgwMiZkaadUgaaaa@390D@ . These i allow to estimate as follows:

    |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@ .

    [0] and the ratio test hence guarantee the convergence of the power series for xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgcMi5kaadggaaaa@3913@ . Now r >r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaafaGaeyOpa4JaamOCaaaa@386B@ can't be true as otherwise we could choose an x with r<|xa|< r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabgYda8iaacYhacaWG4bGaeyOeI0IaamyyaiaacYhacqGH8aapceWGYbGbauaaaaa@3E3B@ , i.e. an x outside the domain of convergence, the series will converge for.

Consider:

  • As convergence is not affected by the first members the above criterion is also valid if the condition a i 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbaabeaakiabgcMi5kaaicdaaaa@39F4@ is only true for almost all i.

  • Another way for calculating r is the Hadamard formula, an implication of the root test [5.9.17]. We note this formula without proof:

    If ( | a n | n ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaakeaabaGaaiiFaiaadggadaWgaaWcbaGaamOBaaqabaGccaGG8baaleaacaWGUbaaaOGaaiykamaaBaaaleaacaWGUbGaeyyzImRaaGimaaqabaaaaa@3F88@ is bounded there is a greatest accumulation point, the so called supremum limit (symbolically lim ¯ | a n | n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaciGGSbGaaiyAaiaac2gaaaWaaOqaaeaacaGG8bGaamyyamaaBaaaleaacaWGUbaabeaakiaacYhaaSqaaiaad6gaaaaaaa@3D67@ ). The radius of convergence now calculates to:

    r={ 1 lim ¯ | a n | n  ,   if   lim ¯ | a n | n 0  ,   if   lim ¯ | a n | n =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2da9maaceaabaqbaeaabiqaaaqaamaalaaabaGaaGymaaqaamaanaaabaGaciiBaiaacMgacaGGTbaaamaakeaabaGaaiiFaiaadggadaWgaaWcbaGaamOBaaqabaGccaGG8baaleaacaWGUbaaaaaakiaabAgacaqGHbGaaeiBaiaabYgacaqGZbWaa0aaaeaaciGGSbGaaiyAaiaac2gaaaWaaOqaaeaacaGG8bGaamyyamaaBaaaleaacaWGUbaabeaakiaacYhaaSqaaiaad6gaaaGccqGHGjsUcaaIWaaabaGaeyOhIuQaaeOzaiaabggacaqGSbGaaeiBaiaabohadaqdaaqaaiGacYgacaGGPbGaaiyBaaaadaGcbaqaaiaacYhacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaaiiFaaWcbaGaamOBaaaakiabg2da9iaaicdaaaaacaGL7baaaaa@6063@
    [5.11.8]

    If ( | a n | n ) n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaakeaabaGaaiiFaiaadggadaWgaaWcbaGaamOBaaqabaGccaGG8baaleaacaWGUbaaaOGaaiykamaaBaaaleaacaWGUbGaeyyzImRaaGimaaqabaaaaa@3F88@ is unbounded we have r=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2da9iaaicdaaaa@3820@ .


 

Proposition and Notation:  Every convergent power series ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ converges (absolutly) for each point x in its domain of convergence. The function

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]

is called its limit function which we denote by the symbol  i=0 a i (Xa) i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaaa@42ED@ .

Proof:  If x]ar,a+r[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaac2facaWGHbGaeyOeI0IaamOCaiaacYcacaWGHbGaey4kaSIaamOCaiaacUfaaaa@3FE3@ , i.e. |xa|<r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgYda8iaadkhaaaa@3C34@ , the number |xa| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaaaa@3A39@ must be exceeded by an element of that set which has r as supremum. Hence ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ will converge for at least one y such that

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

According to [5.11.3]  ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ converges absolutely for x.

Consider:

  • Convergence behaviour at the bounderies of the domain of convergence varies. For example: Employing [5.11.7] we calculate the radius of convergence for the power series ( i=0 n 1 i+1 X i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaSaaaeaacaaIXaaabaGaamyAaiabgUcaRiaaigdaaaGaamiwamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@41F2@ (with expansion point 0) to

    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@ converges for 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaGymaaaa@3711@ (alternating harmonic series [5.9.12]), but not for 1 (harmonic series [5.9.6]).


     
  • If |x|<r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacaGG8bGaeyipaWJaamOCaaaa@3A61@ the number x+a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgUcaRiaadggaaaa@382E@ belongs to the domain of convergence ]ar,a+r[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiyxaiaadggacqGHsislcaWGYbGaaiilaiaadggacqGHRaWkcaWGYbGaai4waaaa@3D62@ . Consequently the series

    ( 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]

    converges absolutely.


     

Example:  

  • ( i=0 n (1) i (X1) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGPbaaaOGaaiikaiaadIfacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@45C3@ is a power series expanded at 1. Due to [5.11.7] the radius of convergence is 1 which results in ]0,2[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiyxaiaaicdacaGGSaGaaGOmaiaacUfaaaa@394F@ as domain of convergence. Considering the limit of the geometric series ([5.9.4]) the limit function turns out to be the restriction of the reciprocal function to ]0,2[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiyxaiaaicdacaGGSaGaaGOmaiaacUfaaaa@394F@  : For any x]0,2[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaac2facaaIWaGaaiilaiaaikdacaGGBbaaaa@3BD0@ we have a distance |1x|<1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaaigdacqGHsislcaWG4bGaaiiFaiabgYda8iaaigdaaaa@3BCD@ , thus:

    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]

     
  • Each of the power series

    ( 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]

    expanded at 0 converges for the whole of MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@36D9@ according to [5.9.18], [5.9.19], [5.9.20]. Their limit functions are exp, sin und cos respectively.


     
  • Let p= i=0 k a i X i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2da9maaqahabaGaamyyamaaBaaaleaacaWGPbaabeaaieaakiaa=HfadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGRbaaniabggHiLdaaaa@4140@ be an arbitrary polynomial. If we define a i 0i>k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbaabeaakiabg2da9iaaicdacaqGSaGaamyAaiabg6da+iaadUgaaaa@3CC8@ , the set of coefficients a 0 ,, a k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIWaaabeaakiaacYcacqWIMaYscaGGSaGaamyyamaaBaaaleaacaWGRbaabeaaaaa@3BC3@ will be extanded to a sequence such that the power series ( i=0 n a i X i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaaieaakiaa=HfadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@40AB@ converges for the whole of MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@36D9@ . The limit function

    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]

    equals  p. So the polynomials are the limit functions of certain power series!

Now we concentrate on the properties of convergent power series. The first idea is to adequately transfer the identity theorem for polynomials (cf. [4.5.*]) to convergent power series: A limit function already equals zero if it has countable many (suitable) zeros!

Proposition (Identity Theorem for Convergent Power Series):  Let  ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ be convergent and ( x n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@38E8@ a sequence in ]ar,a+r[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiyxaiaadggacqGHsislcaWGYbGaaiilaiaadggacqGHRaWkcaWGYbGaai4waaaa@3D62@ such that a x n a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgcMi5kaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsgIRcaWGHbaaaa@3D0F@ . If all sequence members are zeros of the limit function we have:

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

Proof:  As x n a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkziUkaadggaaaa@3A62@ we have a positive number s<r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiabgYda8iaadkhaaaa@385C@ such that x n [as,a+s]  for all  n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGUbaabeaakiabgIGiolaacUfacaWGHbGaeyOeI0Iaam4CaiaacYcacaWGHbGaey4kaSIaam4Caiaac2facaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaaaaa@4DB7@ . The limit  c i=0 | a i | s i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabg2da9maaqahabaGaaiiFaiaadggadaWgaaWcbaGaamyAaaqabaGccaGG8bGaam4CamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaaa@43CA@ exists due to [5.11.10] so that we may estimate for all x with |xa|s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgsMiJkaadohaaaa@3CE6@ and for all  j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgIGiolablwriLcaa@3948@ like this:

| 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]

We now prove [5.11.14] by induction:

  • As all sequence members are zeros of the limit function we get with  j=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabg2da9iaaigdaaaa@3819@ in [1] for all 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@

    Thus the convergence  x n a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkziUkaadggaaaa@3A62@   results in a 0 =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIWaaabeaakiabg2da9iaaicdaaaa@38FF@ .

  • Let a 0 == a k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIWaaabeaakiabg2da9iablAciljabg2da9iaadggadaWgaaWcbaGaam4AaaqabaGccqGH9aqpcaaIWaaaaa@3E39@ be true. We employ [1] with  j=k+2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabg2da9iaadUgacqGHRaWkcaaIYaaaaa@39EC@ and hence get for all 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=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@D43E@

    The equality a k+1 =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaGccqGH9aqpcaaIWaaaaa@3AD2@ is again due to the convergence  x n a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkziUkaadggaaaa@3A62@ .

Consider:

  • [5.11.14] is often used for the so called comparison of coefficients:

    If we have two convergent power series ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ and ( i=0 n b i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamOyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D3@ and if we can find a sequence ( x n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@38E8@ in the intersection of their domains of convergence such that a x n a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgcMi5kaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsgIRcaWGHbaaaa@3D0F@ and 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@ for all n, then both power series already coincide, i.e.

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

    Proof:  The power series ( i=0 n ( a i b i ) (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaaiikaiaadggadaWgaaWcbaGaamyAaaqabaGccqGHsislcaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiykaiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@4823@ (cf. [5.11.17]) satisfies the condition in [5.11.14].
     

Next we show that the convergence of power series is compatible with basic arithmetics.

Proposition:  Let ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ and ( i=0 n b i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamOyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D3@ be two convergent power series and let r 1   and   r 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaaIXaaabeaakiaabwhacaqGUbGaaeizaiaadkhadaWgaaWcbaGaaGOmaaqabaaaaa@3C00@ be their radii of convergence respectively. Then their sum, their difference and their product are convergent power series as well and for |xa|<min{ r 1 , r 2 } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsislcaWGHbGaaiiFaiabgYda8iGac2gacaGGPbGaaiOBaiaacUhacaWGYbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadkhadaWgaaWcbaGaaGOmaaqabaGccaGG9baaaa@4490@ we additionally have:

  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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@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. If b 0 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIWaaabeaakiabgcMi5kaaicdaaaa@39C1@ the quotient ( i=0 n a i (Xa) i ) ( i=0 n b i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaGGOaWaaabCaeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaeaacaGGOaWaaabCaeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaaaaa@524C@ is a convergent power series.
[5.11.19]

Proof:  As series are sequences as well the first three assertions are only variants of the frist three limit theorems [5.6.1] - [5.6.3].

For 4. it is sufficient to represent only the quotient 1 ( i=0 n b i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaiikamaaqahabaGaamOyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaaaa@449E@ as a convergent power series. We begin with the construction of a power series ( i=0 n c i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaam4yamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D4@ such that

( 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]

To that end we need coefficients c i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBaaaleaacaWGPbaabeaaaaa@376B@ which satisfy  j=0 i b j c ij ={ 1 , if  i=0 0 , if  i>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGIbWaaSbaaSqaaiaadQgaaeqaaOGaam4yamaaBaaaleaacaWGPbGaeyOeI0IaamOAaaqabaaabaGaamOAaiabg2da9iaaicdaaeaacaWGPbaaniabggHiLdGccqGH9aqpdaGabaqaauaabaqaceaaaeaacaaIXaGaaeOzaiaabggacaqGSbGaaeiBaiaabohacaWGPbGaeyypa0JaaGimaaqaaiaaicdacaqGMbGaaeyyaiaabYgacaqGSbGaae4CaiaadMgacqGH+aGpcaaIWaaaaaGaay5Eaaaaaa@5372@ .
We define by recursion

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@

and indeed get [2], because:

  • 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@

As b 0 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIWaaabeaakiabgcMi5kaaicdaaaa@39C1@ evaluating the limit function i=0 b i (Xa) i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaaa@42EE@ at the expansion point a will result in a value 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiyIKRaaGimaaaa@37EA@ . We will prove now that this remains true in a complete neighbourhood ]a 1 n ,a+ 1 n [ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiyxaiaadggacqGHsisldaWcaaqaaiaaigdaaeaacaWGUbaaaiaacYcacaWGHbGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGBbaaaa@3EF0@ of a: Suppose for every natural number n> 1 r 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg6da+maalaaabaGaaGymaaqaaiaadkhadaWgaaWcbaGaaGOmaaqabaaaaaaa@3A0E@ there is an x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGUbaabeaaaaa@3785@ with | x n a|< 1 n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGHbGaaiiFaiabgYda8maalaaabaGaaGymaaqaaiaad6gaaaaaaa@3E24@ such that i=0 b i ( x n a) i =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aOGaeyypa0JaaGimaaaa@4601@ . But as a x n a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgcMi5kaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsgIRcaWGHbaaaa@3D0F@   [5.11.14] provides us with  b k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGRbaabeaakiabg2da9iaaicdaaaa@3936@ for all k, a clearly contadicting b 0 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIWaaabeaakiabgcMi5kaaicdaaaa@39C1@ .

So there is a neighbourhood of a with no zeros for the limit function i=0 b i (Xa) i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaaa@42EE@ . For each x in this neighbourhood the forth limit theorem is thus applicable in [2]:

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@ .

So ( i=0 n c i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaam4yamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D4@ converges and due to [2] represents the reciprocal 1 ( i=0 n b i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaiikamaaqahabaGaamOyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaaaa@449E@ .

By definition power series come with a prefixed expansion point. Surprisingly this expansion point could be replaced (locally) with any other point from the domain of convergence. The following theorem plays a technical key role for some of our future work.

Proposition (Rearrangement Theorem):  Let ( i=0 n a i (Xa) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D2@ be any convergent power series and r>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg6da+iaaicdaaaa@3822@ its radius of convergence. Then for every b]ar,a+r[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabgIGiolaac2facaWGHbGaeyOeI0IaamOCaiaacYcacaWGHbGaey4kaSIaamOCaiaacUfaaaa@3FCD@ the following assertion holds: 

There is a convergent power series ( i=0 n b i (Xb) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaamOyamaaBaaaleaacaWGPbaabeaakiaacIcacaWGybGaeyOeI0IaamOyaiaacMcadaahaaWcbeqaaiaadMgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@43D4@ such that

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

Proof:  Please note firstly that s>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg6da+iaaicdaaaa@3823@ . Furthermore we have for all |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]

We use the generalized binomial theorem [5.2.5] to calculate for those x:

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]

So we need to study the double series ( i=0 n j=0 m b ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaacaWGIbWaaSbaaSqaaiaadMgacaaMc8UaamOAaaqabaaabaGaamOAaiabg2da9iaaicdaaeaacaWGTbaaniabggHiLdaaleaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@4702@ , where  b ij { a i (T i j )T (ba) ij (xb) j   , if  ij 0  , if  i<j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGPbGaaGPaVlaadQgaaeqaaOGaeyypa0ZaaiqaaeaafaqaaeGabaaabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcafaqabeGabaaabaGaamyAaaqaaiaadQgaaaGaaiykaiaacIcacaWGIbGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaadMgacqGHsislcaWGQbaaaOGaaiikaiaadIhacqGHsislcaWGIbGaaiykamaaCaaaleqabaGaamOAaaaakiaabAgacaqGHbGaaeiBaiaabYgacaqGZbGaamyAaiabgwMiZkaadQgaaeaacaaIWaGaaeOzaiaabggacaqGSbGaaeiBaiaabohacaWGPbGaeyipaWJaamOAaaaaaiaawUhaaaaa@5E4B@

Due to [3] and according to [5.11.10] the series ( i=0 n | a i | (|ba|+|xb|) i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaGaaiiFaiaadggadaWgaaWcbaGaamyAaaqabaGccaGG8bGaaiikaiaacYhacaWGIbGaeyOeI0IaamyyaiaacYhacqGHRaWkcaGG8bGaamiEaiabgkHiTiaadkgacaGG8bGaaiykamaaCaaaleqabaGaamyAaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@4D8F@ converges. Hence the following estimate holds for all n,m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGTbGaeyicI4SaeSyfHukaaa@3AEE@ :

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@

This proves the boundedness (and consequently the convergence) of the double series ( i=0 n j=0 m | b ij | ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaacaGG8bGaamOyamaaBaaaleaacaWGPbGaaGPaVlaadQgaaeqaaOGaaiiFaaWcbaGaamOAaiabg2da9iaaicdaaeaacaWGTbaaniabggHiLdaaleaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@4917@ . When calculating the limit of the double series ( i=0 n j=0 m b ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaacaWGIbWaaSbaaSqaaiaadMgacaaMc8UaamOAaaqabaaabaGaamOAaiabg2da9iaaicdaaeaacaWGTbaaniabggHiLdaaleaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@4702@ [5.10.26] thus allows us to interchange the order of row limit and column limit (consider also [3] and [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.