<s>[INST] What is the sum of $\sum_{n=1}^\infty\left(\frac{2n+1}{n^4+2n^3+n^2}\right) = $? $$\sum_{n=1}^\infty\left(\frac{2n+1}{n^4+2n^3+n^2}\right)=\sum_{n=1}^\infty\left(\frac{2n+1}{n^2}\frac1{{(n+1)}^2}\right)$$
I assume that I should get a telescoping sum in some way, but I'm couldn't find it yet.
 [/INST] Since
\begin{align*}
\frac{2n+1}{n^4+2n^3+n^2}&=\frac{(n+1)^2-n^2}{n^2(n+1)^2}\\
&=\frac 1{n^2}-\frac 1{(n+1)^2}
\end{align*}
then we have, for an integer $N>1$
$$\sum_{n=1}^N\left(\frac{2n+1}{n^4+2n^3+n^2}\right)=1-\frac1{(N+1)^2}$$
</s>