I'm new here so please let me know if I'm doing something wrong.
I was wondering about the harmonic series especially on ln-series. On wikipedia (https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)) I found something written about it that relate them to the hyperharmonic series. I actually didn't understand exactly the proof of the convergence of this specific series.
It is easy to check that the series $\sum_{n=2} \frac{1}{n(\ln n)^p}$ is a decreasing one. Here the $n$th term is $f(n)=\frac{1}{(n+1)(\ln (n+1))^p}$. Consider the integral $$I= \int_{1}^{\infty} \frac{1}{(x+1)(\ln (x+1))^p} dx$$ Set $\ln (x+1) = t \Rightarrow \frac{dx}{x+1} =dt$, so $I$ becomes, $$I= \int_{\ln 2}^{\infty}\frac{dt}{t^p} =\lim_{n\to \infty} \int_{\ln 2}^{n} \frac{dt}{t^p}$$ It can thus be seen that the integral converges for $p>1$ and diverges for $p\leq 1$. Hope it helps.