The value of 4^ 1/3 .4^ 1/9 .4^ 1/27 ........... is - Vidyadip

Question

The value of ${4^{1/3}}{.4^{1/9}}{.4^{1/27}}...........\infty $ is

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Answer

a
(a) ${4^{1/3}}{.4^{1/9}}{.4^{1/27}}......\infty $

$\therefore $ $S = {4^{1/3 + 1/9 + 1/27}}......\infty $

$ \Rightarrow $ $S = {4^{\left( {\frac{{1/3}}{{1 - 1/3}}} \right)}} = {4^{\frac{{1/3}}{{2/3}}}}$$ $

$\Rightarrow $ $S = {4^{1/2}}$

$ \Rightarrow $ $S = 2$.

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