Calculus (3rd Edition)

Published by W. H. Freeman

Chapter 11 - Infinite Series - Chapter Review Exercises - Page 591: 45

Converges

Work Step by Step

Given $$\sum_{n=2}^{\infty} \frac{n^{2}+1}{n^{3.5}-2}$$ Compare with the convergent series $\displaystyle\sum_{n=2}^{\infty} \frac{1}{n^{1.5}}$ ($p-$series $p>1$), by using the Limit Comparison Test \begin{align*} \lim_{n\to\infty} \frac{a_n}{b_n} &=\lim_{n\to\infty} \frac{n^{3.5}+n^{1.5}}{n^{3.5}-2}\\ &=1 \end{align*} Thus the given series also converges.

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