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Question Number 128124 by shaker last updated on 04/Jan/21

Answered by Dwaipayan Shikari last updated on 04/Jan/21

$$\underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{(1+\frac{1}{n})(1+\frac{2}{n})...(1+\frac{n}{n})}=y$$$$\Rightarrow \frac{1}{n}\underset{k=1}{\sum ^n}log(1+\frac{k}{n})=logy$$$$\Rightarrow {\int }_0^{1}log(1+x)dx=logy\Rightarrow 2log\left(2\right)-1=logy$$$$\Rightarrow y=\frac{4}{e}$$

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