Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 11 - Infinite Series - 11.1 Sequences - Exercises - Page 538: 55


The sequence $y_n$ converges to $0$.

Work Step by Step

We have $$\lim_{n\to \infty}y_n=\lim_{n\to \infty}\frac{e^{n}+(-3)^{n}}{5^{n}}\\ =\lim_{n\to \infty}\frac{e^{n} }{5^{n}}+\frac{ (-3)^{n}}{5^{n}}\\ =\lim_{n\to \infty}(\frac{e }{5} )^{n}+(\frac{-3}{5})^{n}=0.$$ Here we used the facts that $|e/5|\lt 1$ and $|-3/5|\lt 1$. Hence, by Theorem 1, the sequence $y_n$ converges to $0$.
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