Does $\sum_{i=2}^n \frac{3}{n\ln(n)}$ converge or diverge?

Question

I came upon this question while working:

$$\sum_{n=2}^\infty \frac{3}{n\ln(n)}$$

And I was wondering whether it converges or diverges?

A help would be greatly appreciated !

Thank you!

Answer 1

Hint. By the integral test, we have that $\sum_{n\geq 2}\frac{3}{n\ln n}$ converges if and only if $\int_2^{+\infty}\frac{3}{x\ln x}dx$ converges. Use substitution for the integral.

Answer 2

Cauchy condensation test:

$b_n=2^n \dfrac{3}{2^n (n \log 2)}=\left (\dfrac{3}{\log 2}\right )\dfrac{1}{n}$

Hence?

https://en.m.wikipedia.org/wiki/Cauchy_condensation_test