Does $\sum_{n=2}^\infty \frac{\ln(n)}{n (n – 1)}$ converge?
Wolfram alpha suggests that the series converges. But I don't know yet how to prove it.
Attempting to apply the root test I got a complicated limit that I don't know how to evaluate:
$$
\limsup_{n\to\infty}\sqrt[n]{\left|\frac{\ln(n)}{n (n – 1)}\right|},\quad
a_n=\frac{\ln(n)}{n (n – 1)}.
$$
Attempting to apply the ratio test I got
$$
\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_{n}}\right|=1,
$$
which means that the ratio test is inconclusive in this case.
Best Answer
Since$$\lim_{n\to\infty}\frac{\frac{\log n}{n(n-1)}}{\frac{\log n}{n^2}}=1,$$your series converges if and only if $\displaystyle\sum_{n=2}^\infty\frac{\log n}{n^2}$ converges. Which it does, by the integral test. Just use the fact that$$\int_2^\infty\frac{\log x}{x^2}\,\mathrm dx=\lim_{M\to\infty}-\frac1M-\frac{\log M}M+\frac12+\frac{\log2}2=\frac12+\frac{\log2}2.$$