For which $p$'s does $\sum _{n\in \mathbb{N}}\Bigl(\frac{1}{\sqrt{n}\log(1+n)}\Bigr)^{p}$ converge

Question

For which $p's \in \mathbb{R}_\geq1$ does the series $\sum _{n\in \mathbb{N}}\Bigl(\frac{1}{\sqrt{n}\log(1+n)}\Bigr)^{p}$ converge

Thoughts Clearly the sequence $\frac{1}{\sqrt{n}\log(1+n)}$ tends to $0$ as $n$ tends to infinity, although that doesn't mean that much as we know $\frac{1}{n}$ diverges. I've tried a few convergence tests such as the ratio test and found r=1 so that doesn't help, I've tried the integral test but can't see a way to integrate this function so I'm not really sure what to do now.

Answer 1

Hint:

Near $\infty$, the general term is equivalent to $\dfrac1{n^{p/2}\log^pn}$, which is a Bertrand's series.

Now, for the general Bertrand's series $\;\displaystyle\sum_{n=2}^\infty\frac1{n^\alpha (\log n)^\beta}$, it is known that it converges if and only if

• $\alpha>1$, or
• $\alpha=1$ and $\beta>1$.