Question Number 141127 by bobhans last updated on 16/May/21
$$\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{{i}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\:\frac{{n}}{\left({n}+{i}\right)^{\mathrm{2}} }\:=?\: \\ $$
Answered by Ar Brandon last updated on 16/May/21
![L=lim_(n→∞) Σ_(i=0) ^(n−1) (n/((n+i)^2 ))=lim_(n→∞) (1/n)Σ_(i=0) ^(n−1) (1/((1+(i/n))^2 )) =∫_1 ^2 (dx/x^2 )=−[(1/x)]_1 ^2 =1−(1/2)=(1/2)](https://www.tinkutara.com/question/Q141128.png)
$$\mathscr{L}=\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\underset{\mathrm{i}=\mathrm{0}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\frac{\mathrm{n}}{\left(\mathrm{n}+\mathrm{i}\right)^{\mathrm{2}} }=\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{n}}\underset{\mathrm{i}=\mathrm{0}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{1}+\frac{\mathrm{i}}{\mathrm{n}}\right)^{\mathrm{2}} } \\ $$$$\:\:\:\:\:=\int_{\mathrm{1}} ^{\mathrm{2}} \frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{2}} }=−\left[\frac{\mathrm{1}}{\mathrm{x}}\right]_{\mathrm{1}} ^{\mathrm{2}} =\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$