I'm trying to find $\sum\_{k=1}^\infty \frac{1}{k2^k}$. In fact I'm trying to find that $C$ for which $P(\{k\})=\frac{C}{k2^k}$ is a probability measure on $\mathbb{N}$ but presumably the intended solution goes by finding the value of the sum. I want to check if the following is a correct calculation / proof. If I write out the first several terms I don't see a way to make things better.

$$1/2 + 1/8 + 1/24 + 1/64 + ... $$

I thought about the trick used for weakly proving the geometric sum formula: Call the sum $s$, then 

$$s = 1/2 + 1/8 + ... $$
$$4s = 2 + 1/2 + 1/8 + ... $$
so 

$$3s = 2$$

so $s=2/3$.