<s>[INST] Does the series: $\sum_{n=1}^\infty \frac{2n+1}{n^2(n+1)^2}$ converge? does $\sum_{n=1}^\infty \frac{2n+1}{n^2(n+1)^2}$ converge? I think yes, it does, because the $a_n$ in the series converges to zero. but I'm trying to prove this by the help of the fact that:
$\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$
any suggestions? 
 [/INST] $$\frac{2n+1}{n^2(n+1)^2}=\frac{2n}{n^2(n+1)^2}+\frac{1}{n^2(n+1)^2}$$
$$=\frac{2}{n(n+1)^2}+\frac{1}{n^2(n+1)^2}$$
now it is very easy to compare the first term with $\frac{2}{n^3}$ and the second term with $\frac{1}{n^3}$
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