# Chapter 11 - Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Exercises - Page 568: 52

Converges

#### Work Step by Step

Given $$\sum_{n=2}^{\infty} \frac{1}{n^2(\ln n)^{3}}$$ Compare with convergent series $\sum_{n=2}^{\infty} \frac{1}{n^2 }$ ($p-$series $p>1$) , then by using the limit comparison test, we get: \begin{align*} \lim_{n\to \infty} \frac{a_n}{b_n}&=\lim_{n\to \infty} \frac{n^2}{n^2(\ln n)^{3}}\\ &= \lim_{n\to \infty} \frac{1}{ (\ln n)^{3}}\\ &=0 \end{align*} Thus the given series also converges.

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