2^ 1/4 .4^ 1/8 .8^ 1/16 .16^ 1/32 .......... is equal to - Vidyadip

Question

${2^{1/4}}{.4^{1/8}}{.8^{1/16}}{.16^{1/32}}..........$ is equal to

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Answer

b
(b) ${2^{1/4}}{.4^{1/8}}{.8^{1/16}}{.16^{1/32}}.....\infty $

$ = {2^{1/4 + 2/8 + 3/16 + ......}} = {2^S}$, where $S$ is given by

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

$ \Rightarrow $$\frac{1}{2}S = {\rm{ }}\frac{1}{8} + \frac{2}{{16}} + \frac{3}{{32}} + \frac{4}{{64}} + ..........\infty $......$(ii)$

Subtracting $(ii)$ from $(i),$ we get $S = 1$.

Hence required product $ = {2^1} = 2$.

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