For which $p>0$ does $\sum_{n=3}^{\infty}\frac{\log(n)}{n^p}$ converge? For which $p&gt;0$ does $\sum_{n=3}^{\infty}\frac{\log(n)}{n^p}$ converge? I tried all the criteria for series convergence I know, but I'm not getting any further with this exercise. I'm not asking someone to do my homework for me, but could somebody tell me what criteria I should try please or how to proceed?
Consider $3$ cases:

Case #1: $p=1$. From this: $log(n)&gt;1 \implies \dfrac{log(n)}{n} &gt; \dfrac{1}{n} $ you can see by comparison test (with harmonic series) that series diverges.

Case #2: $p&gt;1$. You just use integration test and integrate this $\int_{1}^{+\infty} x^{-p} log(x)dx$ via integration by parts (set $u=log(x)$) and you will see that series converges (because the integral converges).

Case #3: $p&lt;1$. You can easily deduce from previous cases that in this case sum diverges.
As already mentioned elsewhere, this is (apart from the sign and the missing contribution from n=2) the first derivative of the Riemann Zeta function at p.