I'm asked to find the sum of a power series 
for all x in the interior of the domain of convergence, which I found to be [-1,1]. The question also gives the hint to take the second derivative of the series which I found to be 
I understand the idea of differentiating term by term to find the answer but I am unsure of what to do with the (-1)^n
If anyone can help it would be much appreciated.
Deriving term by term, we obtain the series: $$ 2 \sum_{n=1}^{\infty} (-1)^n x^{2n-2} = 2 \sum_{n=0}^{\infty} (-1)^{n+1} x^{2n} = -2 \sum_{n=0}^{\infty} (-x^2)^n $$ This is a geometric series, whose sum is easily obtained. Then, you can evaluate the former series integrating two times (say, over the interval $(0,t)$).