Finding Coefficients of Series Using Integrals and Known Sums

[abc617]

[PLAIN]http://img243.imageshack.us/img243/156/55139700.png

Knowns
Well I know that
\sum \frac{1}{1-x} = 1+x^{2}+x^{3}...

I know the integral is

[PLAIN]http://www4a.wolframalpha.com/Calculate/MSP/MSP32919a6dgiib73geda500000hd72bah4i062h1b?MSPStoreType=image/gif&s=36&w=241&h=38

I've first started with the known sum, then i replaced [x] with [PLAIN]http://www4c.wolframalpha.com/Calculate/MSP/MSP15219a6e2bf832h2gb6000039if18068de4ia05?MSPStoreType=image/gif&s=55&w=20&h=39

\sum \frac{1}{1-(x/2)^2} = 1-(x^2/2)^2-(x^2/2)^3..


Then I tried to integrate the new sum [too long and latek is messing up for me] to get arctan. Then I multiplied it by 16.

I feel that I'm messing up somewhere, I just don't know where.

 

[sutupidmath]

For the integral: In the denominator factor the 4, that is

\frac{16}{4((\frac{x}{2})^2+1)}

and use the hint.