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.

Note that $\log(n)\le \sqrt{n}$.  Hence, we have
$$\left|\frac{\log(n)}{n(n-1)}\right|\le \frac{1}{n^{1/2}(n-1)}$$
Can you finish now?
