How to find the sum of the following power series?

Question

I would like some assistance in summing the following power series. I don't know how to do them myself so therefore I'd like to use this as an example.

$$\sum_{k=2}^{\infty} k(k-1) \cdot x^{k-2}$$

Answer 1

We can see that $R=1$

Let $x\in(-R,R)$

$f\left(x\right)=\sum_{k=0}^{\infty}x^{k}=\frac{1}{1-k}$

$f'\left(x\right)=\left(\sum_{k=0}^{\infty}x^{k}\right)'=\left(\sum_{k=1}^{\infty}kx^{k}\right)=\left(\frac{1}{1-k}\right)'=\frac{k}{\left(1-k\right)^{2}}$

$f''\left(x\right) =\left(\sum_{k=0}^{\infty}x^{k}\right)''=\sum_{k=2}^{\infty}k(k-1)\cdot x^{k-2}=\left(\frac{k}{\left(1-k\right)^{2}}\right)'=\frac{\left(1-k\right)^{2}+k\cdot2\left(1-k\right)}{\left(1-k\right)^{4}}$ $\space\space\space\space\space\space\space\space\space\space=\frac{\left(1-k\right)\left(1-k+2k\right)}{\left(1-k\right)^{4}}=\frac{1+k}{\left(1-k\right)^{4}}$