Question

Using Sandwich theorem, evaluate
$\underset{n\to \infty }{lim}\frac{1}{1+n^2}+\frac{2}{2+n^2}+......+\frac{n}{n+n^2}$

• A. 1
• B. 1/2
• C. 1/4
• D. 1/8

Answer 1

The correct option is B 1/2

for n>1 ,n>r

$n+n^2>r+n^2>n^2$

$\frac{1}{n+n^2}<\frac{1}{1+n^2}<\frac{1}{n^2}$

$\frac{2}{n+n^2}<\frac{2}{2+n^2}<\frac{2}{n^2}$

$\frac{3}{n+n^2}<\frac{3}{3+n^2}<\frac{3}{n^2}$

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Adding on these above equations we get the required equation limit.

LHS of the limit is${\sum }_{r=1}^{\infty }\frac{r}{n+n^2}=\underset{n\to \infty }{lim}\frac{n(n+1)/2}{n+n^2}=\frac{1}{2}$

RHS of the limit is${\sum }_{r=1}^{\infty }\frac{r}{n^2}=\underset{n\to \infty }{lim}\frac{n(n+1)/2}{n^2}=\frac{1}{2}$

Hence by sandwich theorem the value of the limit is $\frac{1}{2}$