I want to find a generating function for $\displaystyle\sum_{r=0}^\infty 2^rx^r = 1 + 2^2x^2 + 2^3x^3 + ...$
I don't think this is a very difficult question but I'm having an issue with it. All I've done in class are simple questions where the coefficient is either a power of $r$ or $(r+1)(r+2)...(r+n)$, a product of consecutive integers.
As far as I can tell, I need to use the exponential generating function where $e^{2x}=\displaystyle\sum_{r=0}^\infty 2^r\frac{x^r}{r!}$. But I have no idea how to deal with the $r!$ part. Is this the kind of approach I want to be taking and if it is, where can I go from here?
If you want the ordinary generating function, you want $\sum_0^\infty 2^n t^n=\sum_0^\infty (2t)^n$. This is the expansion of $\dfrac{1}{1-2t}$. For $1+(2t)+(2t)^2+(2t)^3+\cdots$ is the infinite geometric series with first term $1$ and common ratio $2t$. Note that we have convergence only for $|t|\lt 1/2$.
Remark: For the exponential generating function, we want $\sum_0^\infty \frac{(2t)^n}{n!}$. We recognize this as the Maclaurin series for $e^{2t}$.