Question-144001

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

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

(1+(1/n))^(1/1) (1+(2/n))^(1/2) (1+(3/n))^(1/3) ...=y lim_(n→∞) (1/n)Σ_(k=1) ^n (n/k)log(1+(k/n))=log(y) ⇒∫_0 ^1 ((log(1+x))/x)dx=log(y) ⇒y=exp((π^2 /(12)))

$$\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{\frac{\mathrm{1}}{\mathrm{1}}} \left(\mathrm{1}+\frac{\mathrm{2}}{{n}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \left(\mathrm{1}+\frac{\mathrm{3}}{{n}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} …={y} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{n}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{n}}{{k}}{log}\left(\mathrm{1}+\frac{{k}}{{n}}\right)={log}\left({y}\right) \\ $$$$\Rightarrow\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{log}\left(\mathrm{1}+{x}\right)}{{x}}{dx}={log}\left({y}\right) \\ $$$$\Rightarrow{y}={exp}\left(\frac{\pi^{\mathrm{2}} }{\mathrm{12}}\right) \\ $$

Commented by mnjuly1970 last updated on 20/Jun/21