# Thread: Formal power series: identity problem

1. ## Formal power series: identity problem

Hello,
the problem is to show by calculating that:
$\displaystyle\frac{x(1+x)}{(1-x)^3}=\sum_{i=0}^{\infty}k^2x^k$.

I know the solution has to do something formal power series. I found following pieces: link 1 and link 2 , but even those couldn't get me any further. Also WolframAlpha didn't gave me any clue, though there's some information. I'm completely stuck, which means any help is welcome. Thank you!

2. Originally Posted by Greg98
Hello,
the problem is to show by calculating that:
$\displaystyle\frac{x(1+x)}{(1-x)^3}=\sum_{i=0}^{\infty}k^2x^k$.

I know the solution has to do something formal power series. I found following pieces: link 1 and link 2 , but even those couldn't get me any further. Also WolframAlpha didn't gave me any clue, though there's some information. I'm completely stuck, which means any help is welcome. Thank you!
Very formally we have

$\displaystyle \frac{1}{1-x}=\sum_{k=0}^{\infty}x^k$

Now consider

What is

$\displaystyle x\frac{d}{dx}\left( x\frac{d}{dx}x^k\right)=?$

Why does this help?

3. Originally Posted by Greg98
Hello,
the problem is to show by calculating that:
$\displaystyle\frac{x(1+x)}{(1-x)^3}=\sum_{i=0}^{\infty}k^2x^k$.

I know the solution has to do something formal power series. I found following pieces: link 1 and link 2 , but even those couldn't get me any further. Also WolframAlpha didn't gave me any clue, though there's some information. I'm completely stuck, which means any help is welcome. Thank you!
Proceeding [tediously...] 'step by step' we have...

$\displaystyle \sum_{k=0}^{\infty} k^{2}\ x^{k} = x\ \sum_{k=0}^{\infty} k^{2}\ x^{k-1} = x\ \frac{d}{dx} \sum_{k=0}^{\infty} k\ x^{k}= x\ \frac{d}{dx} (x\ \sum_{k=0}^{\infty} k\ x^{k-1})=$

$\displaystyle = x\ \frac{d}{dx} (x\ \frac{d}{dx} \frac{1}{1-x}) = x\ \frac{d}{dx} \frac{x}{(1-x)^{2}}= x\ \frac{(1-x)^{2} +2 x (1-x)}{(1-x)^{4}} =$

$\displaystyle = x\ \frac{1-x+2 x}{(1-x)^{3}} = x\ \frac{1+x}{(1-x)^{3}}$

Kind regards

$\chi$ $\sigma$