I have the following limit
$$\lim \limits_{n \to \infty}\left({1 \over 2^1}+{3 \over 2^2} + {5 \over 2^3} + \dots + {2n-1 \over 2^n}\right)$$
My idea is to separate it into geometric progressions and rewrite them using the equation for the sum of first terms of a geometric progression. I manage to get one geometric progression but the other elements of the sum don't follow a geometric pattern. Do you have any hints as to how I might separate the terms? Or, alternatively, if there is a simpler way to do this?
