University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 9 - Section 9.1 - Sequences - Exercises - Page 488: 74


Converges to $0$

Work Step by Step

Consider $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \dfrac{(\dfrac{10}{11})^n}{(\dfrac{9}{10})^n+(\dfrac{11}{12})^n}$ Since, $\lim\limits_{n \to \infty}x^n=0$ So, $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \dfrac{(\dfrac{10}{11})^n}{(\dfrac{9}{10})^n+(\dfrac{11}{12})^n}= \lim\limits_{n \to \infty} \dfrac{(\dfrac{12}{11} \cdot \dfrac{10}{11})^n}{(\dfrac{12}{11} \cdot \dfrac{9}{10})^n+(\dfrac{12}{11} \cdot \dfrac{11}{12})^n}=\dfrac{0}{0+1}$ or, $=0$ Hence, $\lim\limits_{n \to \infty} a_n=0$ and {$a_n$} is convergent and converges to $0$
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