Question

Value of sum ${\sum }_{k=1}^{\infty }{\sum }_{n=1}^{\infty }\frac{k}{2^{n+k}}=$

Answer 1

${\sum }_{k=1}^{\infty }\frac{k}{2^k}{\sum }_{n=1}^{\infty }\frac{1}{2^n}$

$={\sum }_{k=1}^{\infty }\frac{k}{2^k}(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}...\infty )$

= ${\sum }_{k=1}^{\infty }\frac{k}{2^k}=\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...\infty$

Let $S=\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...\infty$ ... (1)
$\frac{1}{2}S=\frac{1}{2^2}+\frac{2}{2^3}+...\infty$ ... (2)

Subtracting (2) from (1), we get,

$\frac{1}{2}S=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...\infty$

$\Rightarrow \frac{1}{2}S=1$

$\Rightarrow S=2$