How to compute this series

Question

I have to compute this series :

$$\sum_{k=0}^\infty (k+1)x^{2k}$$

First, I have $$|x|<1$$ but then I don't know how to begin ...

Answer 1

Use the fact that this is a power series with radius of convergence $r=1$. Thus, for $\lvert y\rvert < r$, $$ \sum_{k=0}^\infty (k+1)y^k = \frac{d}{dy} \sum_{k=0}^\infty y^{k+1} = \frac{d}{dy} \sum_{k=1}^\infty y^{k} $$ and $$ \sum_{k=1}^\infty y^{k} = \frac{y}{1-y}. $$ Now, you can compute the derivative of $\frac{y}{1-y}$, and evaluate it on $x^2$ (which has $x^2 < r$ indeed).