Question

Prove that :

$(2^{\frac{1}{4}}{.4}^{\frac{2}{8}}{.8}^{\frac{3}{16}}{.16}^{\frac{4}{32}}....\infty )=2.$

Answer 1

LHS = $2^{\frac{1}{4}}{.4}^{\frac{1}{8}}{.8}^{\frac{1}{16}}{.16}^{\frac{1}{32}}....\infty =2^{(\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}.\infty )}=2^{(\frac{1}{2^2}+\frac{2}{2^3}+\frac{3}{2^4}+\frac{4}{2^5}+....\infty )}=2^{\frac{1}{2^2}\{1+\frac{2}{2}+\frac{3}{2^2}+\frac{4}{2^3}...\infty \}}=2^{\frac{1}{2^2}\{\frac{1}{1-\frac{1}{2}}+\frac{1.\frac{1}{2}}{{(1-\frac{1}{2})}^2}\}}=2^{2^{\frac{1}{2}\{2+2\}}}=2^1=2$ = RHS