Proving that this series has a finite sum Consider the following series
$$\sum_{n=1}^{\infty}\dfrac{\log n}{n(n-1)}$$
I have tried to use the ratio test, but then I would get
$$\dfrac{(n-1)\log(n+1)}{(n+1)\log n}$$
And taking the limit as $n \to \infty$ would yield 1 so I don't think it would help.

Hint. The Ratio Test is inconclusive here. Note that if $a\in(1,2)$ then
$$\lim_{n\to \infty}\frac{\dfrac{\log n}{n(n-1)}}{\dfrac{1}{n^{a}}}=\lim_{n\to \infty}\dfrac{n^a\log n}{n(n-1)}=0$$
which implies that there is an $N>0$ such that for all $n\geq N$,
$$0\leq \dfrac{\log n}{n(n-1)}\leq \dfrac{1}{n^{a}}.$$
What may we conclude?
