Question Number 203059 by hassanmpsy last updated on 08/Jan/24
Commented by witcher3 last updated on 11/Jan/24
![U_n =Σ_(k=1) ^n ((n(1+(k/n)))/(n^2 (2+2(k/n)+((k/n))^2 )))=(1/n)Σ_(k=1) ^n f((k/n)) lim_(n→∞) U_n =∫_0 ^1 f(x)dx=∫_0 ^1 ((1+x)/(x^2 +2x+2))=(1/2)∫_0 ^1 ((2x+2)/(x^2 +2x+2))dx =(1/2)[ln(x^2 +2x+2)]_0 ^1 =(1/2)ln(3)](https://www.tinkutara.com/question/Q203165.png)
$$\mathrm{U}_{\mathrm{n}} =\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{n}\left(\mathrm{1}+\frac{\mathrm{k}}{\mathrm{n}}\right)}{\mathrm{n}^{\mathrm{2}} \left(\mathrm{2}+\mathrm{2}\frac{\mathrm{k}}{\mathrm{n}}+\left(\frac{\mathrm{k}}{\mathrm{n}}\right)^{\mathrm{2}} \right)}=\frac{\mathrm{1}}{\mathrm{n}}\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{f}\left(\frac{\mathrm{k}}{\mathrm{n}}\right) \\ $$$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}+\mathrm{x}}{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{2x}+\mathrm{2}}{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{2}}\mathrm{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{ln}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{2}\right)\right]_{\mathrm{0}} ^{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\left(\mathrm{3}\right) \\ $$