Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Exercises - Page 568: 16



Work Step by Step

Given $$\sum_{n=1}^{\infty} \frac{1 }{(2n)!}$$ By using the Ratio test, we get: \begin{align*} \rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\ &=\lim _{n \rightarrow \infty} \frac{(2n)! }{(2n+2)!} \\ &=\lim _{n \rightarrow \infty} \frac{(2n)! }{(2n+2)(2n+1)(2n)!} \\ &=\lim _{n \rightarrow \infty} \frac{1}{(2n+2)(2n+1)} \\ &=0<1 \end{align*} Thus the series converges.
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