So there is this question being asked:
For $X\sim \mathrm{Geom}(p)$, find $E(2^x)$ (if it is finite) and $E(2^{-x})$ (if it is finite).
I know how to find expectation, but my doubt is regarding how to determine if expectation will be finite or not before calculating it (as is being asked by this particular question.)
$$\operatorname{E[2^X]}=\sum_{k=1}^{\infty}{2^k(1-p)^{k-1}p}$$ The sum converges if $p>1/2$ and diverges if $p\leqslant 1/2$ $$\operatorname{E[2^{-X}]}=\sum_{k=1}^{\infty}{2^{-k}(1-p)^{k-1}p}$$ The sum converges for all $0\leqslant p\leqslant 1$