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Question Number 144001 by mnjuly1970 last updated on 20/Jun/21

Answered by Dwaipayan Shikari last updated on 20/Jun/21

$${(1+\frac{1}{n})}^{\frac{1}{1}}{(1+\frac{2}{n})}^{\frac{1}{2}}{(1+\frac{3}{n})}^{\frac{1}{3}}...=y$$$$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\underset{k=1}{\sum ^n}\frac{n}{k}log(1+\frac{k}{n})=log\left(y\right)$$$$\Rightarrow {\int }_0^1\frac{log(1+x)}{x}dx=log\left(y\right)$$$$\Rightarrow y=exp\left(\frac{{\pi }^2}{12}\right)$$

Commented by mnjuly1970 last updated on 20/Jun/21

$$thanksalotmrpayan...$$

Answered by abdullahoudou last updated on 20/Jun/21

$$e^{{\pi }^2/12}$$

Commented by mnjuly1970 last updated on 20/Jun/21

$$tashakormaster..$$

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