Question

Prove that $2^{1/4}\cdot 4^{1/8}\cdot 8^{1/16}\cdot {16}^{1/32}\cdot ........\cdot \infty =2$

Answer 1

$2^{1/4}\cdot 4^{1/8}\cdot 8^{1/16}\cdot {16}^{1/32}\cdot ........\cdot \infty =2$

$2^{1/4}\cdot 4^{1/8}\cdot 8^{1/16}\cdot {16}^{1/32}\cdot ........\cdot \infty =2^{1/4}\cdot 2^{2/8}\cdot 2^{3/16}\cdot 2^{4/32}\cdot ........\cdot \infty$

$=2^{1/4+2/8+3/16+4/32+....\infty }$

Let $M=\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}+....\infty$

$\frac{M}{2}=\frac{1}{8}+\frac{2}{16}+\frac{3}{32}+....\infty$

$M-\frac{M}{2}=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+....\infty$

$\frac{M}{2}=\frac{1/4}{1-1/2}$

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

$\therefore 2^{1/4}\cdot 4^{1/8}\cdot 8^{1/16}\cdot {16}^{1/32}\cdot ........\cdot \infty =2^M$

$=2$