Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 10: Infinite Sequences and Series - Section 10.6 - Alternating Series and Conditional Convergence - Exercises 10.6 - Page 603: 34


Absolutely Convergent

Work Step by Step

A series $ \Sigma a_n$ is defined as the absolutely convergent when the series $ \Sigma |a_n|$ is convergent. From the series, we notice that $ \Sigma_{n=1}^{\infty} |\dfrac{(-1)^{n-1}}{n^2+2n+1}|= \Sigma_{n=2}^{\infty} \dfrac{1}{n^2}$ This suggests a $P-$ series with common ratio $r=2 \gt 1$ Remember that a p- series is said to be convergent when the common ratio $r \gt 1$. This implies that the series is Absolutely Convergent.
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