# Chapter 10: Infinite Sequences and Series - Practice Exercises - Page 636: 32

Diverges

#### Work Step by Step

Let $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} \dfrac{\ln n}{\ln (\ln n)}$ But, $\lim\limits_{n \to \infty} \dfrac{\ln n}{\ln (\ln n)}=\dfrac{\infty}{\infty}$ Apply L-hospital's rule such that $\lim\limits_{x \to k}\dfrac{p(x)}{q(x)}=\lim\limits_{x \to k}\dfrac{p'(x)}{q'(x)}$ $\implies \lim\limits_{n \to \infty}\dfrac{n \ln (n)}{n}=\infty$ Hence, the series diverges.

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