Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - 11.3 Convergence of Series with Positive Terms - Exercises - Page 557: 56


The series $\sum_{n=1}^{\infty} \frac{1}{3^{n^2}} $ converges.

Work Step by Step

We compare the given series $\sum_{n=1}^{\infty} \frac{1}{3^{n^2}} $ with the geometric series $\sum_{n=1}^{\infty} \frac{1}{3^{n}} $ which is a convergent series ($r=\frac{1}{3}\lt 1$). Thus, by the comparison test, the given series $\sum_{n=1}^{\infty} \frac{1}{3^{n^2}} $ also converges.
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