Convergence of the series $\sum_{n=1}^\infty \ln(n) \ln\left(1+ \frac{1}{n(n+2)}\right)$ I need your help to show that the series $\sum_{n=1}^\infty \ln(n)\ln\left(1+ \frac{1}{n(n+2)}\right)$ converges. I have tried to use the ratio and root test, but in both cases the limit is $1$. Could you help me, please?
Thanks in advance.

Since $$
\ln (1 + x) \le x
$$ for every $x>0$ you have that
$$
\ln \left( {1 + \frac{1}{{n\left( {n + 2} \right)}}} \right) \le \frac{1}{{n\left( {n + 2} \right)}}
$$ for every $n \in \mathbb N$.
Now$$
\ln (n)\ln \left( {1 + \frac{1}{{n\left( {n + 2} \right)}}} \right) \le \frac{{\ln n}}{{n\left( {n + 2} \right)}}
$$
But $
\ln n < \sqrt n 
$ for every $n\geq 1$ thus
$$
\ln (n)\ln \left( {1 + \frac{1}{{n\left( {n + 2} \right)}}} \right) \le \frac{1}{{\sqrt n \left( {n + 2} \right)}} \le \frac{1}{{n^{{\textstyle{3 \over 2}}} }}
$$
and the thesis follows by direct comparison test.
