Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - Chapter Review Exercises - Page 592: 81



Work Step by Step

Given $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}+\sqrt{n+1}}$$ This is an alternating series. Consider $ a_{n}=\dfrac{1}{\sqrt{n}+\sqrt{n+1}}$, since $\{a_n \}$ is a decreasing sequence and \begin{align*} \lim _{n \rightarrow \infty} a_{n}&=\lim _{n \rightarrow \infty} \dfrac{1}{\sqrt{n}+\sqrt{n+1}}\\ &=0 \end{align*} Then by using the Leibniz Test, the given series converges.
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