A fair coin is tossed until it lands on heads. The player receives 2^n dollars with n being the number of tails before the heads, i.e. 1$ if first toss is H, 2 if 2nd, 4 if 3rd and so on. What is the expectation for how much money the player ends up?
The expectation for number of tails hit is 1, but obviously you can't just take 2^(E(# of tails)). Running a quick simulation also revealed this doesn't converge to 2.
On the other hand, taking $$E = \sum_{n=1}^\infty2^{n-1}(1-p)^{n-1}p = \sum_{n=1}^\infty2^{n-1}\cdot(\frac{1}{2})^{n-1}\cdot\frac{1}{2}=\infty$$ is just senseless...
What am I doing wrong?