Question

Evaluate $\underset{n\to \infty }{lim}(\frac{1}{n^2+1}+\frac{2}{n^2+2}+\frac{3}{n^2+3}+.......+\frac{n}{n^2+n})$

Answer 1

Solution :- given,

$=\underset{n\to \infty }{lim}(\frac{1}{n^2+1}+\frac{2}{n^2+2}+\frac{3}{n^2+3}+...+\frac{n}{n^n+n})$

$=\underset{n\to \infty }{lim}[\frac{1}{n^2+(1+\frac{1}{n^2})}+\frac{2}{n^2(1+\frac{1}{n^2})}+....+\frac{n}{n^2(1+\frac{n}{n^2})}]$

$=\underset{n\to \infty }{lim}\frac{1}{n^2}[\frac{1}{1+\frac{1}{n^2}}+\frac{2}{1+\frac{2}{n^2}}+.....+\frac{n}{1+\frac{n}{n^2}}]$

$=0\times \underset{n\to \infty }{lim}[\frac{1}{1+\frac{1}{n^2}}+\frac{2}{1+\frac{2}{n^2}}+...\frac{n}{1+\frac{n}{n^2}}]$

$=0$ (Ans)