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 591: 34


See the proof below.

Work Step by Step

We have $$\lim _{n \to \infty}a_n=\lim _{n \to \infty} \left(b-\tan ^{-1} n^{2}\right)=b-\tan^{-1}\infty\\ =b-\frac{\pi}{2}$$ Whenever $b\neq \pi/2$, then $\lim _{n \to \infty}a_n\neq0$ and hence the series diverges (it converges only if $\lim _{n \to \infty}a_n=0$).
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