Does $\Sigma_{n=1}^{\infty} \dfrac {\sqrt {2n-1}~ \log (4n+1)} {n(n+1)}$ converge?
Attempt: I have been trying to use the comparison test for a while, but I can't find a suitable comparator.
For Example, since, $\Sigma_{n=1}^{\infty} \dfrac {1}{n(n+1)}$ converges, I was thinking of an expression involving the above to use as a comparator. But can not find one.
Please guide me on how to move ahead.
Thank you very much for your help.