Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - 11.4 Absolute and Conditional Convergence - Exercises - Page 563: 25


The series $\sum_{n=1}^{\infty} \frac{3^n+(-2)^n}{5^{n}}$ converges.

Work Step by Step

In the series $\sum_{n=1}^{\infty} \frac{3^n+(-2)^n}{5^{n}}$, we have the geometric series $\sum_{n=1}^{\infty} (\frac{3 }{5})^n$, which converges since $r=3/5\lt 1$. For the series $\sum_{n=1}^{\infty} \frac{(-2)^n}{5^{n}}$, the positive term $b_n=(\frac{2}{5})^n$; then we have $$\lim_{n\to \infty}b_n=\lim_{n\to \infty}(\frac{2}{5})^n=0$$ since $2/5\lt 1$. Using the alternating series test, the series $\sum_{n=1}^{\infty} \frac{(-2)^n}{5^{n}}$ converges. Hence the series $\sum_{n=1}^{\infty} \frac{3^n+(-2)^n}{5^{n}}$ converges.
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