Suma szeregu

[Molniya]

Proszę o pomoc z obliczeniem szeregu:

\(\displaystyle{ \sum_{n=1}^{ \infty } an^{2}x ^{n}}\)

[octahedron]

\(\displaystyle{ \sum_{n=1}^{ \infty } an^{2}x ^{n}=ax\sum_{n=1}^{ \infty } n^{2}x ^{n-1}=ax\sum_{n=1}^{ \infty } (nx ^{n})'=ax\left(\sum_{n=1}^{ \infty } nx ^{n}\right)'=ax\left(x\sum_{n=1}^{ \infty } nx ^{n-1}\right)'=\\=ax\left(x\sum_{n=1}^{ \infty } (x ^{n})'\right)'=ax\left(x\left(\sum_{n=1}^{ \infty } x ^{n}\right)'\right)'=ax\left(x\left(\frac{x}{1-x}\right)'\right)'=ax\left(\frac{x}{(1-x)^2}\right)'=\\=\frac{ax(1+x)}{(1-x)^3}}\)

[Molniya]

Dzięki!!!