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: 8


The series $\sum_{n=1}^{\infty} \frac{(-1)^{n} }{ 1+(1/n)}$ diverges.

Work Step by Step

We have the absolute series $$\sum_{n=1}^{\infty} |\frac{(-1)^{n} }{ 1+(1/n)}|=\sum_{n=1}^{\infty} \frac{1}{ 1+(1/n)}.$$ So, we have $$\lim_{n\to \infty }b_n=\lim_{n\to \infty } \frac{1}{ 1+(1/n)}=\lim_{n\to \infty } \frac{1 }{ 1+0}=1\neq 0.$$ Thus the positive series diverges by the divergence test. We see that the terms $\frac{1}{ 1+(1/n)}$ are not decreasing. Thus the original series $\sum_{n=1}^{\infty} \frac{(-1)^{n} }{ 1+(1/n)}$ does not converge absolutely or conditionally.
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