A mean that doesn't exist

Hello everyone, I've came across this problem and I need to see if my approach is right...

An urn initially contains one Black and one White ball. At each stage, a ball is randomly chosen and then replaced along with another of the same color. Let $\displaystyle X$ denote the selection number of the 1st Black ball chosen.

(a)Find the pdf of $\displaystyle X$.

(b)Show that the $\displaystyle E(X)$ doesn't exist for $\displaystyle X$.

for (a), I thought that if I came accross a black ball I'll replace it so

$\displaystyle P(X=1)=\frac{1}{2},\ P(X=2)=(\frac{1}{2})(\frac{1}{2})$ and for any $\displaystyle k,\ P(X=k)={(\frac{1}{2})}^k$.

As for (b), $\displaystyle E(X)=\sum_k k{(\frac{1}{2})}^k$ which is 2!!

How could this be?!

Any help is very much appreciated!