TITLE: Prove the series $\sum_{n=2}^\infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$ converges
QUESTION [5 upvotes]: I need to prove the series $$\sum\limits_{n=2}^\infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$$ converges.
I cant use comparison test, since if I take away any term in denominator I know it will diverge. Any help please?

REPLY [6 votes]: Hint. Since the general term is a positive and decreasing sequence, by the integral test, your series behaves as the following integral:
$$
\int_3^{\infty}\frac{{\rm d} x}{x \ln x \:(\ln(\ln x))^2}. \tag1
$$ By the change of variable $$u=\ln(\ln x), \quad du=\dfrac{dx}{x \ln x },$$ we easily have

$$
\begin{align}
\int_3^{\infty}\frac{{\rm d} x}{x \ln x \:(\ln(\ln x))^2} \mathrm dx=\int_{\ln(\ln 3)}^{\infty}\frac{{\rm d} u}{u^2}
=\left[ -\frac{1}{u}\right]_{\ln(\ln 3)}^{\infty}
=\frac1{\ln(\ln 3)}.
\end{align}
$$

Your initial series is convergent.
Edit. Thanks to user109899's remark, we have singled out the term coming from $n=2$.