Question

$\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\frac{5}{32}+\dots =$

Answer 1

Let $S=\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\frac{5}{32}+\dots$
$\frac{S}{2}=\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}+\frac{5}{64}+\dots$

On subtracting, we get
$S-\frac{S}{2}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\dots \dots$
$\Rightarrow \frac{S}{2}=\frac{\frac{1}{2}}{1-\frac{1}{2}}=1$
$\Rightarrow S=2$