Find $E\left[\frac{1}{1+X}\right]$ if $X$ is a geometric random variable with parameter $p$

Question

Let $X$ be a geometric random variable with parameter $p$. Find the expectation of $\frac{1}{1+X}$.

I understand that $\sum\frac{1}{1+x}p(x)...$ then $\sum\frac{1}{1+x}p(1-p)^{x-1}$. I need help simplifying the series.

Answer 1

Hint

You can solve your exercise using (and understanding)

$$\sum_{n=1}^\infty nx^{n-1}=\frac{\mathrm d }{\mathrm d x}\sum_{n=0}^\infty x^{n}=\frac{\mathrm d }{\mathrm d x}\left(\frac{1}{1-x}\right).$$