Problem with the correction of series exercise Well, I have a little problem with the correction of an exercise, I have to calculate :
$$
S(x)=\sum_{n=2}^{\infty} \frac{n+(-1)^{n+1}}{n+(-1)^n}x^n, x \in \mathbb{R}
$$
So I have :
$$
S(x)=\sum_{p=1}^{\infty} x^{2p} - 2\sum_{p=1}^{\infty} \frac{x^{2p}}{2p+1} + \sum_{p=1}^{\infty}x^{2p+1} + \sum_{p=1}^{\infty} \frac{x^{2p+1}}{p}
$$
It's the same on the correction but my problem is after, they simplify this four sums like this :
$$
-1-x+\sum_{n=1}^{\infty} x^n - \frac{2}{x}\sum_{p=1}^{\infty}\frac{x^{2p+1}}{2p+1}+x\sum_{p=1}^{\infty} \frac{(x^2)^p}{p}
$$
And it maybe be ridiculous for you but I don't understand why :
$$
\sum_{p=1}^{\infty}x^{2p+1} + \sum_{p=1}^{\infty} \frac{x^{2p+1}}{p}=x\sum_{p=1}^{\infty} \frac{(x^2)^p}{p}
$$
For me :
$$\sum_{p=1}^{\infty}x^{2p+1} + \sum_{p=1}^{\infty} \frac{x^{2p+1}}{p} = x*\sum_{p=1}^{\infty} \left((x^2)^p \left(1+\frac{1}{p}\right)\right)
$$
Thank's before for your help !
Shadock 
 A: The first problem is to assume that the 2 latter terms yield the result you have trouble understanding. First, I believe there may be a little typo. Allow me to demonstrate. You state:
$$S(x)=\sum_{p=1}^{\infty} x^{2p} - 2\sum_{p=1}^{\infty} \frac{x^{2p}}{2p+1} + \sum_{p=1}^{\infty}x^{2p+1} + \sum_{p=1}^{\infty} \frac{x^{2p+1}}{p}$$
I am going to work with the first and third terms. Clearly
$$\sum_{p=1}^{\infty} x^{2p} + \sum_{p=1}^{\infty}x^{2p+1} = \sum_{p=2}^{\infty}x^{p}$$
Furthermore,
$$\sum_{p=2}^{\infty}x^{p} = -1 - x +\sum_{p=0}^{\infty}x^{p}$$
So we have combined the first and third terms together. Going back to our original expression,
$$S(x) =  -1 - x +\sum_{p=0}^{\infty}x^{p} - 2\sum_{p=1}^{\infty} \frac{x^{2p}}{2p+1} + \sum_{p=1}^{\infty} \frac{x^{2p+1}}{p}$$
$$= -1 - x +\sum_{p=0}^{\infty}x^{p} -\frac{2}{x}\sum_{p=1}^{\infty} \frac{x^{2p+1}}{2p+1} + x\sum_{p=1}^{\infty} \frac{(x^{2})^{p}}{p}. $$
So the only difference is in the limits of the result of the 3rd term (there lies the typo, I believe). Hope this helps! Be sure to exhaust all possibilities before coming to conclusions :) !
