Processing math: 5%
 

5.7. Monotone und beschränkte Folgen


Im letzten Abschnitt konnten wir bequem und schnell über das Grenzwertverhalten von Folgen entscheiden. Die Grenzwertsätze lieferten die nötigen Techniken. Allerdings ist der Bereich der Folgen, die mit dieser Methode bearbeitet werden können, deutlich eingeschränkt, denn die Folgen müssen ja eine bestimmte Struktur aufweisen. Es ist daher sinnvoll, nach weiteren Konvergenztests zu suchen.

Wir greifen noch einmal die Eigenschaften monoton und beschränkt auf. Beide haben allein keinen bzw. nur einen geringen Bezug zur Konvergenz. Ihre Kombination aber ist überraschenderweise sehr günstig und liefert ein oft benutztes Konvergenzkriterium. Im Unterschied zu den Grenzwertsätzen allerdings gibt es keine Auskunft über den Grenzwert selbst. Außerdem ist seine Gültigkeit streng an die reellen Zahlen gebunden; auf Folgen in etwa, ist es nicht anwendbar (siehe [5.7.11]).

Satz:  Für jede reelle Zahlenfolge (an) gilt:

Ist (an) monoton und beschränkt, so ist (an) auch konvergent.
[5.7.1]

Beweis:  Wir führen den Beweis für eine monoton wachsende Folge. Zunächst gibt es auf Grund der Beschränktheit ein s, so dass

ans  für alle  n.

Die Menge der Folgenglieder {an|n} ist also eine nicht-leere, nach oben beschränkte Teilmenge von . Sie besitzt - und hier geht die Besonderheit der reellen Zahlen ein - nach dem Vollständigkeitsaxiom eine kleinste obere Schranke, das Supremum also. Wir setzen nun  g  und zeigen:

a n g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgkziUkaadEgaaaa@3A51@

Sei dazu ein ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@38D2@ vorgegeben. Da g die kleinste obere Schranke von { a n |n } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadggadaWgaaWcbaGaamOBaaqabaGccaGG8bGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaOGaaiyFaaaa@3F81@ ist, kann gε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabgkHiTiabew7aLbaa@38E9@ keine obere Schranke mehr sein. Es muß also mindestens ein Folgenglied oberhalb von gε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabgkHiTiabew7aLbaa@38E9@ liegen, d.h. es gibt ein n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIWaaabeaakiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3B58@ , so dass a n 0 >gε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbWaaSbaaWqaaiaaicdaaeqaaaWcbeaakiabg6da+iaadEgacqGHsislcqaH1oqzaaa@3CF2@ . Beachtet man, dass g auch eine gewöhnliche obere Schranke der monoton wachsenden Folge ( a n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3951@ ist, so erhält man für alle n n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@39FB@ :

gε< a n 0 a n g<g+ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabgkHiTiabew7aLjabgYda8iaadggadaWgaaWcbaGaamOBamaaBaaameaacaaIWaaabeaaaSqabaGccqGHKjYOcaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyizImQaam4zaiabgYda8iaadEgacqGHRaWkcqaH1oqzaaa@47CC@ ,

also  a n ]gε,g+ε[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgIGiolaac2facaWGNbGaeyOeI0IaeqyTduMaaiilaiaadEgacqGHRaWkcqaH1oqzcaGGBbaaaa@4261@ . Gemäß [5.4.2] ist dies die Behauptung.

Bei der Anwendung des neuen Kriteriums "monoton und beschränkt" gehen wir stets in zwei Schritten vor: Zunächst erhalten wir die reine Konvergenzaussage, anschließend ermitteln wir den Grenzwert selbst.

In einem ersten Beispiel studieren wir Folgen des Typs ( q n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadghadaahaaWcbeqaaiaad6gaaaGccaGGPaaaaa@38E2@ .

Bemerkung:  Für q MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaiabgIGiolabl2riHcaa@3953@ hat man:

|q|<1 q n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadghacaGG8bGaeyipaWJaaGymaiabgkDiElaadghadaahaaWcbeqaaiaad6gaaaGccqGHsgIRcaaIWaaaaa@4142@

[5.7.2]

|q|>1( q n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadghacaGG8bGaeyOpa4JaaGymaiabgkDiElaacIcacaWGXbWaaWbaaSqabeaacaWGUbaaaOGaaiykaaaa@3FF8@   ist divergent

[5.7.3]

Beweis:  
1.  

Nach [5.5.6] reicht es | q n |=|q | n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadghadaahaaWcbeqaaiaad6gaaaGccaGG8bGaeyypa0JaaiiFaiaadghacaGG8bWaaWbaaSqabeaacaWGUbaaaOGaeyOKH4QaaGimaaaa@4156@ zu zeigen, so dass wir o.E. 0q<1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgsMiJkaadghacqGH8aapcaaIXaaaaa@3A8D@ annehmen dürfen. Wir multiplizieren diese Ungleichung mit q n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaCaaaleqabaGaamOBaaaaaaa@377F@ und erhalten

0 q n+1 q n <1  für alle  n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgsMiJkaadghadaahaaWcbeqaaiaad6gacqGHRaWkcaaIXaaaaOGaeyizImQaamyCamaaCaaaleqabaGaamOBaaaakiabgYda8iaaigdacaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaaaaa@4DD2@ .

( q n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadghadaahaaWcbeqaaiaad6gaaaGccaGGPaaaaa@38E2@ ist also monoton fallend und beschränkt, somit konvergent, etwa gegen g.

Bleibt zu zeigen g=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2da9iaaicdaaaa@3815@ . Dazu betrachten wir zusätzlich die Folge ( q n+1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadghadaahaaWcbeqaaiaad6gacqGHRaWkcaaIXaaaaOGaaiykaaaa@3A7F@ , die ebenfalls gegen g konvergiert, und benutzen einen kleinen Trick, indem wir den Grenzwert von ( q n+1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadghadaahaaWcbeqaaiaad6gacqGHRaWkcaaIXaaaaOGaaiykaaaa@3A7F@ über den dritten Grenzwertsatz ein zweites Mal berechnen. Also:

q n+1 g q n+1 =q q n qg MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaaqaaiaadghadaahaaWcbeqaaiaad6gacqGHRaWkcaaIXaaaaOGaeyOKH4Qaam4zaaqaaiaadghadaahaaWcbeqaaiaad6gacqGHRaWkcaaIXaaaaOGaeyypa0JaamyCaiabgwSixlaadghadaahaaWcbeqaaiaad6gaaaGccqGHsgIRcaWGXbGaeyyXICTaam4zaaaaaaa@4C47@

Da aber ( q n+1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadghadaahaaWcbeqaaiaad6gacqGHRaWkcaaIXaaaaOGaaiykaaaa@3A7F@ genau einen Grenzwert hat, muss  g=qgg(1q)=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2da9iaadghacqGHflY1caWGNbGaeyi1HSTaam4zaiabgwSixlaacIcacaaIXaGaeyOeI0IaamyCaiaacMcacqGH9aqpcaaIWaaaaa@46D0@   gelten, also g=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2da9iaaicdaaaa@3815@ , denn nach Voraussetzung ist q1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaiabgcMi5kaaigdaaaa@38E1@ .

2.  

Für |q|>1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadghacaGG8bGaeyOpa4JaaGymaaaa@3A22@ , etwa |q|=1+x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadghacaGG8bGaeyypa0JaaGymaiabgUcaRiaadIhaaaa@3BFF@   mit einem x>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg6da+iaaicdaaaa@3828@ , folgt mit der Bernoullischen Ungleichung

| q n |=|q | n = (1+x) n 1+nx. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadghadaahaaWcbeqaaiaad6gaaaGccaGG8bGaeyypa0JaaiiFaiaadghacaGG8bWaaWbaaSqabeaacaWGUbaaaOGaeyypa0JaaiikaiaaigdacqGHRaWkcaWG4bGaaiykamaaCaaaleqabaGaamOBaaaakiabgwMiZkaaigdacqGHRaWkcaWGUbGaamiEaaaa@4A25@

Mit (1+nx) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacqGHRaWkcaWGUbGaamiEaiaacMcaaaa@3A4F@ ist daher auch ( q n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadghadaahaaWcbeqaaiaad6gaaaGccaGGPaaaaa@38E2@ unbeschränkt, also divergent.

Beachte:

  • Der Fall |q|=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadghacaGG8bGaeyypa0JaaGymaaaa@3A20@ ist bereits bekannt: ( (1) n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaamOBaaaakiaacMcaaaa@3AED@ ist divergent und 1 n =11 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymamaaCaaaleqabaGaamOBaaaakiabg2da9iaaigdacqGHsgIRcaaIXaaaaa@3BB7@ .
     

Wir setzen das Kriterium erneut ein, um den bekannten Konvergenzen 1 n k 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamOBamaaCaaaleqabaGaam4AaaaaaaGccqGHsgIRcaaIWaaaaa@3AF5@ weitere hinzuzufügen.

Bemerkung:  Für r,s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaacYcacaWGZbGaeyicI4SaeSyfHu6aaWbaaSqabeaacqGHxiIkaaaaaa@3C14@ und  q r s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaiabg2da9maalaaabaGaamOCaaqaaiaadohaaaaaaa@3964@   gilt:

1 n s 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaWaaOqaaeaacaWGUbaaleaacaWGZbaaaaaakiabgkziUkaaicdaaaa@3AEB@

[5.7.4]

1 n q 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamOBamaaCaaaleqabaGaamyCaaaaaaGccqGHsgIRcaaIWaaaaa@3AFB@

[5.7.5]

Beweis:  
1.  

Die Monotonie des Wurzeloperators liefert für alle n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3A68@ die folgenden Ungleichungen:

1nn+1 1 n s n+1 s 1 1 n s 1 n+1 s 0. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaaqaaaqaaiaaigdacqGHKjYOcaWGUbGaeyizImQaamOBaiabgUcaRiaaigdaaeaacqGHshI3aeaacaaIXaGaeyizIm6aaOqaaeaacaWGUbaaleaacaWGZbaaaOGaeyizIm6aaOqaaeaacaWGUbGaey4kaSIaaGymaaWcbaGaam4CaaaaaOqaaiabgkDiEdqaaiaaigdacqGHLjYSdaWcaaqaaiaaigdaaeaadaGcbaqaaiaad6gaaSqaaiaadohaaaaaaOGaeyyzIm7aaSaaaeaacaaIXaaabaWaaOqaaeaacaWGUbGaey4kaSIaaGymaaWcbaGaam4CaaaaaaGccqGHLjYScaaIWaaaaaaa@59D9@

( 1 n s ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaalaaabaGaaGymaaqaamaakeaabaGaamOBaaWcbaGaam4CaaaaaaGccaGGPaaaaa@399D@ ist also monoton fallend und beschränkt, somit konvergent, etwa gegen g. Zur Ermittlung von g nutzen wir den dritten Grenzwertsatz:

1 n = ( 1 n s ) s g s g s =0g=0. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamOBaaaacqGH9aqpcaGGOaWaaSaaaeaacaaIXaaabaWaaOqaaeaacaWGUbaaleaacaWGZbaaaaaakiaacMcadaahaaWcbeqaaiaadohaaaGccqGHsgIRcaWGNbWaaWbaaSqabeaacaWGZbaaaOGaeyO0H4Taam4zamaaCaaaleqabaGaam4Caaaakiabg2da9iaaicdacqGHshI3caWGNbGaeyypa0JaaGimaaaa@4CD9@
2.  

Wir setzen wieder den dritten Grenzwertsatz ein und benutzen das Ergebnis aus 1.:

1 n q = 1 n r s = ( 1 n s ) r 0 r =0. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamOBamaaCaaaleqabaGaamyCaaaaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaWGUbWaaWbaaSqabeaadaWcaaqaaiaadkhaaeaacaWGZbaaaaaaaaGccqGH9aqpcaGGOaWaaSaaaeaacaaIXaaabaWaaOqaaeaacaWGUbaaleaacaWGZbaaaaaakiaacMcadaahaaWcbeqaaiaadkhaaaGccqGHsgIRcaaIWaWaaWbaaSqabeaacaWGYbaaaOGaeyypa0JaaGimaaaa@494B@

Das nächsten Beispiel ist klassisch. Es führt uns zu einer der wichtigsten mathematischen Konstanten, der sog. Eulerschen Zahl.

Beispiel:  

  • ( (1+ 1 n ) n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaaiykaaaa@3CA0@ und  ( (1+ 1 n ) n+1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaakiaacMcaaaa@3E3D@ sind konvergent.
[5.7.6]

Beweis:  Beide Konvergenzaussagen lassen sich am besten gleichzeitig beweisen. Wir zeigen in drei Schritten dass beide Folgen monoton und beschränkt sind:
1.  

( (1+ 1 n ) n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaaiykaaaa@3CA0@ ist monoton wachsend, denn mit der Bernoullischen Ungleichung erhalten wir für n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3A68@ zunächst

(1+ 1 n+1 ) n+1 (1 1 n+1 ) n+1 = (1 1 (n+1) 2 ) n+1 1 1 n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbGaey4kaSIaaGymaaaacaGGPaWaaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaakiabgwSixlaacIcacaaIXaGaeyOeI0YaaSaaaeaacaaIXaaabaGaamOBaiabgUcaRiaaigdaaaGaaiykamaaCaaaleqabaGaamOBaiabgUcaRiaaigdaaaGccqGH9aqpcaGGOaGaaGymaiabgkHiTmaalaaabaGaaGymaaqaaiaacIcacaWGUbGaey4kaSIaaGymaiaacMcadaahaaWcbeqaaiaaikdaaaaaaOGaaiykamaaCaaaleqabaGaamOBaiabgUcaRiaaigdaaaGccqGHLjYScaaIXaGaeyOeI0YaaSaaaeaacaaIXaaabaGaamOBaiabgUcaRiaaigdaaaaaaa@5D2C@

und daraus:

(1+ 1 n+1 ) n+1 1 (1 1 n+1 ) n = ( n+1 n ) n = (1+ 1 n ) n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbGaey4kaSIaaGymaaaacaGGPaWaaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaakiabgwMiZoaalaaabaGaaGymaaqaaiaacIcacaaIXaGaeyOeI0YaaSaaaeaacaaIXaaabaGaamOBaiabgUcaRiaaigdaaaGaaiykamaaCaaaleqabaGaamOBaaaaaaGccqGH9aqpcaGGOaWaaSaaaeaacaWGUbGaey4kaSIaaGymaaqaaiaad6gaaaGaaiykamaaCaaaleqabaGaamOBaaaakiabg2da9iaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbaaaaaa@568E@ .
 
2.  

( (1+ 1 n ) n+1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaakiaacMcaaaa@3E3D@ ist monoton fallend: Wir setzen noch einmal die Bernoullische Ungleichung ein und erhalten für n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3A68@

( (n+1) 2 (n+1) 2 1 ) n+1 = (1+ 1 (n+1) 2 1 ) n+1 (1+ 1 (n+1) 2 ) n+1 1+ 1 n+1 . MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaalaaabaGaaiikaiaad6gacqGHRaWkcaaIXaGaaiykamaaCaaaleqabaGaaGOmaaaaaOqaaiaacIcacaWGUbGaey4kaSIaaGymaiaacMcadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIXaaaaiaacMcadaahaaWcbeqaaiaad6gacqGHRaWkcaaIXaaaaOGaeyypa0JaaiikaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaGGOaGaamOBaiabgUcaRiaaigdacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGymaaaacaGGPaWaaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaakiabgwMiZkaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaaiikaiaad6gacqGHRaWkcaaIXaGaaiykamaaCaaaleqabaGaaGOmaaaaaaGccaGGPaWaaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaakiabgwMiZkaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbGaey4kaSIaaGymaaaaaaa@66F3@

Damit können wir jetzt folgendermaßen abschätzen:

(1+ 1 n+1 ) n+2 ( (n+1) 2 (n+1) 2 1 ) n+1 (1+ 1 n+1 ) n+1 = ( (n+1) 2 (n+2)n n+2 n+1 ) n+1 = ( n+1 n ) n+1 = (1+ 1 n ) n+1 . MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@82D8@
 
3.  

( (1+ 1 n ) n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaaiykaaaa@3CA0@ und  ( (1+ 1 n ) n+1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaakiaacMcaaaa@3E3D@ sind beschränkt, denn da (1+ 1 1 ) 1 =2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIXaaaaiaacMcadaahaaWcbeqaaiaaigdaaaGccqGH9aqpcaaIYaaaaa@3C99@ und (1+ 1 1 ) 2 =4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIXaaaaiaacMcadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaI0aaaaa@3C9C@ folgt aus dem gerade gezeigten Monotonieverhalten:

2 (1+ 1 n ) n (1+ 1 n ) n+1 4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgsMiJkaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaeyizImQaaiikaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbaaaiaacMcadaahaaWcbeqaaiaad6gacqGHRaWkcaaIXaaaaOGaeyizImQaaGinaaaa@495B@ .

Beachte:

  • Da (1+ 1 n ) n+1 (1+ 1 n ) n = (1+ 1 n ) n 1 n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbaaaiaacMcadaahaaWcbeqaaiaad6gacqGHRaWkcaaIXaaaaOGaeyOeI0IaaiikaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbaaaiaacMcadaahaaWcbeqaaiaad6gaaaGccqGH9aqpcaGGOaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaad6gaaaGaaiykamaaCaaaleqabaGaamOBaaaakiabgwSixpaalaaabaGaaGymaaqaaiaad6gaaaGaeyOKH4QaaGimaaaa@5142@ besitzen ( (1+ 1 n ) n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaaiykaaaa@3CA0@ und  ( (1+ 1 n ) n+1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaakiaacMcaaaa@3E3D@ über [5.7.6] hinaus sogar denselben Grenzwert! Die Zahl

    elim (1+ 1 n ) n =lim (1+ 1 n ) n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaiabg2da9iGacYgacaGGPbGaaiyBaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaeyypa0JaciiBaiaacMgacaGGTbGaaiikaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbaaaiaacMcadaahaaWcbeqaaiaad6gacqGHRaWkcaaIXaaaaaaa@4B4E@

    [5.7.7]

    heißt die Eulersche Zahl. Wir werden ihr noch oft begegnen und neben [5.7.7] weitere Berechnungsmöglichkeiten finden. Leonhard Euler selbst berechnet in seinem 1748 veröffentlichten Werk Introductio in Analysin infinitorum bereits die ersten 18 Stellen der Eulerschen Zahl: 

    e = 2.718281828459045235....

    Dabei hat Euler bei seiner Berechnung sicherlich nicht die Darstellung in [5.7.7] benutzt, denn wie man im folgenden Applet selbst ausprobieren kann, konvergiert z.B. die Folge ( (1+ 1 n ) n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaaiykaaaa@3CA0@ äußerst langsam. So sichert etwa

    (Das Applet arbeitet mit sog. constructive real numbers und kann theoretisch Zahlen mit beliebig vielen Dezimalstellen darstellen. Die Voreinstellung von 100 Stellen läßt sich zwar hier auf
    z.B.   ,
    man beachte aber, dass die benutzte Rechnerkonfiguration Grenzen setzt. Bei mehr als 3000 Nachkommastellen dürften sich erste Probleme einstellen!)

    Im übernächsten Abschnitt geben wir eine Folge an, die deutlich schneller gegen e konvergiert. Mit ihrer Hilfe gelingt auch der Nachweis, dass die Eulersche Zahl irrational ist.


     

Über die Beschränktheit von ( (1+ 1 n ) n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaaiykaaaa@3CA0@ , etwa nach oben durch 4, gewinnen wir die Konvergenz einer weiteren, nicht elementaren Folge.

Beispiel:  

  • n n 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWGUbaaleaacaWGUbaaaOGaeyOKH4QaaGymaaaa@3A1C@
[5.7.8]

Beweis:  Da (1+ 1 n ) n 4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbaaaiaacMcadaahaaWcbeqaaiaad6gaaaGccqGHKjYOcaaI0aaaaa@3DBA@ , gilt für alle n4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgwMiZkaaisdaaaa@38E0@ der Reihe nach:

(1+ 1 n ) n n n+1n n n = n n+1 n n+1 n+1 n n . MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabmqaaaqaaiaacIcacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaeyizImQaamOBaaqaaiaad6gacqGHRaWkcaaIXaGaeyizImQaamOBaiabgwSixpaakeaabaGaamOBaaWcbaGaamOBaaaakiabg2da9maakeaabaGaamOBamaaCaaaleqabaGaamOBaiabgUcaRiaaigdaaaaabaGaamOBaaaaaOqaamaakeaabaGaamOBaiabgUcaRiaaigdaaSqaaiaad6gacqGHRaWkcaaIXaaaaOGaeyizIm6aaOqaaeaacaWGUbaaleaacaWGUbaaaaaaaaa@5649@

( n n ) n4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaakeaabaGaamOBaaWcbaGaamOBaaaakiaacMcadaWgaaWcbaGaamOBaiabgwMiZkaaisdaaeqaaaaa@3C70@ ist also monoton fallend und wegen 1 n n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgsMiJoaakeaabaGaamOBaaWcbaGaamOBaaaaaaa@39DA@ auch beschränkt, und somit konvergent etwa gegen g1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabgwMiZkaaigdaaaa@38D6@ . Dabei können wir den Fall g>1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg6da+iaaigdaaaa@3818@ ausschließen, denn da MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@36D9@ archimedisch geordnet ist, finden wir zu jedem x>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg6da+iaaicdaaaa@3828@ ein n 4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaeyyzImRaaGinaaaaaaa@3BFD@ , so dass 1<(n1) x 2 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgYda8iaacIcacaWGUbGaeyOeI0IaaGymaiaacMcacqGHflY1daWcaaqaaiaadIhadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaaaaaaa@4022@ . Das allgemeine Binomialtheorem [5.2.5] ermöglicht nun für dieses n die folgende Abschätzung:

n<n(n1) x 2 2 =(T n 2 )T 1 n2 x 2 i=0 n (T n i )T 1 ni x i = (1+x) n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6893@ .

Also hat man n n <1+x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWGUbaaleaacaWGUbaaaOGaeyipaWJaaGymaiabgUcaRiaadIhaaaa@3B12@ , und damit 1+xg MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgUcaRiaadIhacqGHGjsUcaWGNbaaaa@3AB6@ , denn man weiß, dass n n g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWGUbaaleaacaWGUbaaaOGaeyyzImRaam4zaaaa@3A26@ ist.

Die Konvergenz von ( n n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaakeaabaGaamOBaaWcbaGaamOBaaaakiaacMcaaaa@38CD@ zieht weitere Konvergenzen nach sich. Für 1an MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgsMiJkaadggacqGHKjYOcaWGUbaaaa@3B67@ hat man 1 a n n n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgsMiJoaakeaabaGaamyyaaWcbaGaamOBaaaakiabgsMiJoaakeaabaGaamOBaaWcbaGaamOBaaaaaaa@3D8D@ , so dass aus dem Schachtelsatz [5.5.8] folgt

a n 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWGHbaaleaacaWGUbaaaOGaeyOKH4QaaGymaaaa@3A0F@
[5.7.9]

Dies gilt dann auch für 0<a<1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgYda8iaadggacqGH8aapcaaIXaaaaa@39CC@ a n = 1 1 a n 1 1 =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWGHbaaleaacaWGUbaaaOGaeyypa0ZaaSaaaeaacaaIXaaabaWaaOqaaeaadaWcaaqaaiaaigdaaeaacaWGHbaaaaWcbaGaamOBaaaaaaGccqGHsgIRdaWcaaqaaiaaigdaaeaacaaIXaaaaiabg2da9iaaigdaaaa@4135@ .
 

Mit weiteren Beispielen zeigen wir die Bedeutung des Kriteriums "monoton und konvergent" für die Untersuchung rekursiver Folgen auf.

Beispiel:  

  • Für die durch  a 1 2 a n+1 2 a n a n +1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIXaaabeaakiabg2da9iaaikdacqGHNis2caWGHbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabg2da9maalaaabaGaaGOmaiaadggadaWgaaWcbaGaamOBaaqabaaakeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaey4kaSIaaGymaaaaaaa@45E9@   rekursiv gegebene Folge ( a n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3951@ gilt:  a n 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgkziUkaaigdaaaa@3A20@ .

Beweis:  Wir zeigen zunächst per Induktion, und damit ist ( a n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3951@ bereits beschränkt,
 

1 a n 2  für alle  n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laaigdacqGHKjYOcaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyizImQaaGOmaiaabAgacaqG8dGaaeOCaiaabccacaqGHbGaaeiBaiaabYgacaqGLbGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@4D64@

►   122, also:  1 a 1 2. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laaigdacqGHKjYOcaaIYaGaeyizImQaaGOmaiaabYcacaqGGaGaaeyyaiaabYgacaqGZbGaae4BaiaabQdacaaIXaGaeyizImQaamyyamaaBaaaleaacaaIXaaabeaakiabgsMiJkaaikdaaaa@4BEA@

►   Aus 1 a n 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgsMiJkaadggadaWgaaWcbaGaamOBaaqabaGccqGHKjYOcaaIYaaaaa@3F78@ erhält man die Abschätzung

1= 2 a n a n + a n 2 a n a n +1 22 1+1 =2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laaigdacqGH9aqpdaWcaaqaaiaaikdacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgUcaRiaadggadaWgaaWcbaGaamOBaaqabaaaaOGaeyizIm6aaSaaaeaacaaIYaGaamyyamaaBaaaleaacaWGUbaabeaaaOqaaiaadggadaWgaaWcbaGaamOBaaqabaGccqGHRaWkcaaIXaaaaiabgsMiJoaalaaabaGaaGOmaiabgwSixlaaikdaaeaacaaIXaGaey4kaSIaaGymaaaacqGH9aqpcaaIYaaaaa@5544@

und hat damit  1 a n+1 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laaigdacqGHKjYOcaWGHbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabgsMiJkaaikdaaaa@4258@ .

Ferner ist ( a n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3951@ monoton fallend, denn mit a n 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadggadaWgaaWcbaGaamOBaaqabaGccqGHLjYScaaIXaaaaa@3E5B@ hat man auch a n ( a n 1)0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadggadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamyyamaaBaaaleaacaWGUbaabeaakiabgkHiTiaaigdacaGGPaGaeyyzImRaaGimaaaa@436A@ , so dass mit der folgenden Äquivalenz die Behauptung bewiesen ist:

a n a n+1 a n 2 a n a n +1 a n 2 + a n 2 a n a n 2 a n 0 a n ( a n 1)0. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=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@7165@

Insgesamt ist daher ( a n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3951@ konvergent, etwa gegen g[1,2] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadEgacqGHiiIZcaGGBbGaaGymaiaacYcacaaIYaGaaiyxaaaa@4022@ . g ist aber auch Grenzwert von ( a n+1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaaiykaaaa@3AEE@ , so dass wir g auch mit Hilfe der Grenzwertsätze berechnen können:

g a n+1 = 2 a n a n +1 2g g+1 . MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadEgacqGHqgcRcaWGHbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabg2da9maalaaabaGaaGOmaiaadggadaWgaaWcbaGaamOBaaqabaaakeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaey4kaSIaaGymaaaacqGHsgIRdaWcaaqaaiaaikdacaWGNbaabaGaam4zaiabgUcaRiaaigdaaaaaaa@4E07@

Grenzwerte sind eindeutig, also hat man die Gleichung  g= 2g g+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadEgacqGH9aqpdaWcaaqaaiaaikdacaWGNbaabaGaam4zaiabgUcaRiaaigdaaaaaaa@3FFE@   und damit (beachte g[1,2] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadEgacqGHiiIZcaGGBbGaaGymaiaacYcacaaIYaGaaiyxaaaa@4022@ ):

g 2 +g=2gg(g1)=0g=1. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadEgadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGNbGaeyypa0JaaGOmaiaadEgacqGHuhY2caWGNbGaaiikaiaadEgacqGHsislcaaIXaGaaiykaiabg2da9iaaicdacqGHuhY2caWGNbGaeyypa0JaaGymaaaa@4E24@

Das letzte Beispiel ist ein sehr nützliches Hilfsmittel zur approximativen Berechnung von Quadratwurzeln. Wie der Name verrät, ist dies ein sehr altes Verfahren.

Bemerkung (Babylonisches Wurzelziehen):  Ist a>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadggacqGH+aGpcaaIWaaaaa@3C73@ , so ist für jeden Startwert a 1 >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadggadaWgaaWcbaGaaGymaaqabaGccqGH+aGpcaaIWaaaaa@3D64@ die durch die Rekursion  a n+1 1 2 ( a n + a a n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOaGaamyyamaaBaaaleaacaWGUbaabeaakiabgUcaRmaalaaabaGaamyyaaqaaiaadggadaWgaaWcbaGaamOBaaqabaaaaOGaaiykaaaa@4753@   gegebene Folge ( a n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3951@ konvergent, genauer:

a n a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadggadaWgaaWcbaGaamOBaaqabaGccqGHsgIRdaGcaaqaaiaadggaaSqabaaaaa@3EC8@
[5.7.10]

Beweis:  Zunächst vergewissern wir uns durch eine induktive Überlegung, dass a n >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadggadaWgaaWcbaGaamOBaaqabaGccqGH+aGpcaaIWaaaaa@3D9C@ für alle n. Da Quadrate stets positiv sind, liefert uns der Schluss
 

0 ( a n a a n ) 2 = a n 2 a + a a n 2 a a n + a a n a 1 2 ( a n + a a n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=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@682B@

die Abschätzung a n+1 a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaeyyzIm7aaOaaaeaacaWGHbaaleqaaaaa@403E@ . Für alle n2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laad6gacqGHLjYScaaIYaaaaa@3D40@ gilt daher:

a n a n+1   = a n a n 2 a 2 a n = a n 2 a 2 a n a 2 a 2 a = a 2 a 2 =0. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=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@634A@

Also ist ( a n+1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laacIcacaWGHbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiaacMcaaaa@3ED0@ monoton fallend und wegen  a a n+1 a 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=paakaaabaGaamyyaaWcbeaakiabgsMiJkaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaeyizImQaamyyamaaBaaaleaacaaIYaaabeaaaaa@43BA@   auch beschränkt, insgesamt daher konvergent, etwa gegen g a >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadEgacqGHLjYSdaGcaaqaaiaadggaaSqabaGccqGH+aGpcaaIWaaaaa@3F4A@ . Da auch a n g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadggadaWgaaWcbaGaamOBaaqabaGccqGHsgIRcaWGNbaaaa@3EB3@ können wir zur Ermittlung von g wieder unseren "Standardtrick" einsetzen: Aus

g a n+1 = 1 2 ( a n + a a n ) g0 1 2 (g+ a g ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadEgacqGHqgcRcaWGHbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccqGHRaWkdaWcaaqaaiaadggaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaaakiaacMcadaWfqaqaaiabgkziUcWcbaGaam4zaiabgcMi5kaaicdaaeqaaOWaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOaGaam4zaiabgUcaRmaalaaabaGaamyyaaqaaiaadEgaaaGaaiykaaaa@5655@

erhalten wir dabei  g= 1 2 (g+ a g )2g=g+ a g g 2 =ag= a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadEgacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiaacIcacaWGNbGaey4kaSYaaSaaaeaacaWGHbaabaGaam4zaaaacaGGPaGaeyi1HSTaaGOmaiaadEgacqGH9aqpcaWGNbGaey4kaSYaaSaaaeaacaWGHbaabaGaam4zaaaacqGHuhY2caWGNbWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaamyyaiabgsDiBlaadEgacqGH9aqpdaGcaaqaaiaadggaaSqabaaaaa@567D@ .

Beachte:

  • Das Grundprinzip des Babylonischen Wurzelziehens ist einfach und genial zugleich. Die Äquivalenz

    a n < a a a n > a a = a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadggadaWgaaWcbaGaamOBaaqabaGccqGH8aapdaGcaaqaaiaadggaaSqabaGccqGHuhY2daWcaaqaaiaadggaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaaakiabg6da+maalaaabaGaamyyaaqaamaakaaabaGaamyyaaWcbeaaaaGccqGH9aqpdaGcaaqaaiaadggaaSqabaaaaa@485A@

    lesen wir so: Ist a n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadggadaWgaaWcbaGaamOBaaqabaaaaa@3BD0@ zu klein (zu groß), so ist a a n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=paalaaabqaq=laadggaaeaba9VaamyyamaaBaaaleaacaWGUbaabeaaaaaaaa@3F4C@ zu groß (zu klein). In jedem Fall ist daher das arithmetische Mittel a n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=laadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaaa@3D6D@ ein vermutlich besser Approximationswert.
     

  • Ist a rational, so zeigt ein einfacher Induktionsbeweis, dass ( a n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3951@ eine Folge in MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=lablQriKcaa@3B3B@ ist. Insbesondere hat man also:

    Es gibt eine monotone und beschränkte Folge ( a n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3951@ in MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=lablQriKcaa@3B3B@ , die gegen die irrationale Zahl 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=paakaaabqaq=laaikdaaSqabaaaaa@3BE5@ konvergiert.

    [5.7.11]

    Da ( a n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3951@ nur einen Limes besitzen kann, ist ( a n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3951@ in MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepe0xh9as0=LqLs=Jarpepeea0=as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbqaq=lablQriKcaa@3B3B@ divergent!
     

  • Das Babylonische Wurzelziehen ist ein recht schnelles Approximationsverfahren. Selbst bei ungüstigen Startwerten reichen oft nur wenige Schritte, um bereits die ersten zehn Dezimalstellen zu sichern. Wir zeigen dies am Beispiel der Wurzel aus . Dazu wählen wir den Anfangswert a 1 = MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIXaaabeaakiabg2da9aaa@3846@  , legen die Anzahl der Iterationsschritte auf fest und starten dann die Iteration:

    n= MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9aaa@3762@   a n = MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabg2da9aaa@387E@  


5.6. 5.8.