1. ## [Solved] Power Series problem

Hi,
This math problem is really confusing me, any ideas on how to start?

i have to explicitely compute the function f defined by

f(x)=$\displaystyle \sum_{n=1}^{\infty} n^2x^_2n}$

as well as find its domain.

from wolframalpha i know the correct function is $\displaystyle f(x)=\frac{-x^2(x^2+1)}{(x^2-1)^3}$ although i have no idea how to prove this and it seems quite complex for a power series.

Any help is greatly appreciated.

2. ## Re: Power Series problem

$\displaystyle t=x^2 (say)$
$\displaystyle \sum_{n=0}^{\infty} t^n = \frac{1}{1-t}$
Differentiate wrt t,
$\displaystyle \sum_{n=0}^{\infty} nt^{n-1} = \frac{1}{(1-t)^2}$
$\displaystyle \sum_{n=0}^{\infty} nt^{n} = \frac{t}{(1-t)^2}$
Differentiate wrt t,
$\displaystyle \sum_{n=0}^{\infty} n^2t^{n-1} = \frac{1+t}{(1-t)^3}$
$\displaystyle \sum_{n=0}^{\infty} n^2t^{n} = \frac{t(1+t)}{(1-t)^3}$
For n=0,the term itself is 0.So,
$\displaystyle \sum_{n=1}^{\infty} n^2t^{n} = \frac{t(1+t)}{(1-t)^3}$

Now Substitute back $\displaystyle t=x^2$.
To look for the domain check in which region your series converges.

Thanks alot!