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Question Number 12828 by 786786AM last updated on 03/May/17

$$\mathrm{The}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{infinite}\mathrm{product}$$$$\sqrt{3}\cdot \sqrt[4]{9}\cdot \sqrt[8]{27}\cdot \sqrt[16]{81}...\mathrm{to}\mathrm{\infty }\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}.$$

Answered by prakash jain last updated on 04/May/17

$$3^{1/2}\cdot 3^{2/4}\cdot 3^{3/8}...=3^{(\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\frac{4}{2^4}+...)}=3^S$$$$S=\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\frac{4}{2^4}+...\left(i\right)$$$$\frac{S}{2}=\frac{1}{2^2}+\frac{2}{2^3}+\frac{3}{2^4}+....\left(ii\right)$$$$subtracting\left(ii\right)from\left(i\right)$$$$\frac{S}{2}=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...=\frac{1/2}{1-1/2}=1$$$$S=2$$$$3^S=3^2=9$$

Commented by mrW1 last updated on 04/May/17

$$verynicetrick!$$

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