Let $X$ be a discrete random variable with probability $P(X=x)=\frac{1}{2^x}$ for $x=1,2,3,\dots$. Then the expected value of $X$ is:

Expected value $E(X)=\sum xP(X=x)={\sum }_{x=1}^{\infty }\frac{x}{2^x}$.Let $S=\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\dots$ This is an Arithmetico-Geometric Progression (AGP).$S=\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\dots$$\frac{1}{2}S=\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\dots$ Subtracting the two:$S-\frac{1}{2}S=\frac{1}{2}+(\frac{2}{4}-\frac{1}{4})+(\frac{3}{8}-\frac{2}{8})+\dots$$\frac{1}{2}S=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots$ The RHS is an infinite GP with $a=1/2,r=1/2$.Sum $=\frac{1/2}{1-1/2}=1$.$\frac{1}{2}S=1⟹S=2$.