<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] Notice that
$$ \frac{1}{n^2} - \frac{1}{(n+1)^2} = \frac{n^2+2n+1-n^2}{n^2(n+1)^2}, $$
so the series telescopes:
$$ \sum_{n=1}^m \frac{2n+1}{n^2(n+1)^2} = 1-\frac{1}{2^2} + \frac{1}{2^2} -\dotsb - \frac{1}{m^2} + \frac{1}{m^2} - \frac{1}{(m+1)^2} = 1- \frac{1}{(m+1)^2} \to 1 $$
as $m \to \infty$
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