A Theorem say the $\displaystyle p$-series $\displaystyle \sum_{n=2}^\infty \frac {1}{n^p}$ converges if $\displaystyle p>1$ and diverges otherwise.

The question says "Find the positive values of $\displaystyle p$ for which the series $\displaystyle \sum_{n=2}^\infty \frac 1{n (\ln n)^p}$ converges.

The answer is $\displaystyle p=1$ (from back of book). There is no explanation of how they got it and I have none either.How do you find that?

I thought maybe I would compare $\displaystyle \frac 1{n (\ln n)^p}$ to $\displaystyle \frac {1}{n^p}$. If $\displaystyle \frac 1{(n \ln n)^p} < \frac {1}{n^p}$, then the former converges, too. For $\displaystyle p>1$, and since $\displaystyle n>1$, then $\displaystyle (\ln n)^p>1 $and thus $\displaystyle \frac 1{(n \ln n)^p} < \frac {1}{n^p}$. How am I doing so far?

But what about when $\displaystyle p<1$? Maybe for some values close to, but less than 1, $\displaystyle \sum_{n=2}^\infty \frac 1{n (\ln n)^p}$ still converges. How do I find out?