Finding the coffecient for a generating function

Question

Let $f(x) = {2 \over {1-2x}}$.

How can you tell the expansion is: $\{2x + 2^2x^2 + 2^3x^3 + ...\}$

I am familiar with the geometric series and I guess there must be a connection between the two.

Answer 1

$$f(x)=\frac{2}{1-2x}=2\sum_{k=0}^\infty 2^k x^k=2+2^2x+2^3x^2+\cdots$$