<s>[INST] Sum of the power series $\sum\limits_{n=2}^{\infty}\frac{n^2}{(n-1)(n+2)}x^n$ The series is convergent for $|x|<1$ and divergent for $|x|>1$.
I can't find the sum. Integrating three times gives
$$\frac{n^2}{(n^2-1)(n+3)(n+2)^2}x^{n+3}$$
that should have a closed form.
How to find the sum of this series?
 [/INST] You can write
$$
\frac{n^2}{(n-1)(n+2)}=1+\frac{-n+2}{(n-1)(n+2)}=
1+\frac{1}{3}\frac{1}{n-1}-\frac{4}{3}\frac{1}{n+2}
$$
with partial fraction decomposition. So you reduce the problem to finding the sum of
$$
\sum_{n=2}^{\infty}x^n,
\quad
\sum_{n=2}^{\infty}\frac{x^n}{n-1},
\quad
\sum_{n=2}^{\infty}\frac{x^n}{n+2}
$$
</s>