Question

Solve:$\underset{n\to \infty }{lim}\frac{1}{1+n^2}+\frac{2}{2+n^2}+......+\frac{n}{n+n^2}$

• A. $1$
• B. $\frac{1}{2}$
• C. $2$
• D. $\frac{2}{3}$

Answer 1

Answer: (B)

$\frac{1}{2}$

$\frac{1}{1+n^2}+\frac{2}{2+n^2}+\cdots +\frac{n}{n+n^2}<\frac{1}{1+n^2}+\frac{2}{1+n^2}+\cdots +\frac{n}{1+n^2}$
$⟹\frac{1}{1+n^2}+\frac{2}{2+n^2}+\cdots +\frac{n}{n+n^2}<\frac{1+2+\dots n}{1+n^2}$
$⟹\frac{1}{1+n^2}+\frac{2}{2+n^2}+\cdots +\frac{n}{n+n^2}<\frac{n(n+1)}{2(1+n^2)}$
Also
$\frac{1}{n+n^2}+\frac{2}{n+n^2}+\cdots +\frac{n}{n+n^2}<\frac{1}{1+n^2}+\frac{2}{2+n^2}+\cdots +\frac{n}{n+n^2}$
$⟹\frac{1+2+\dots n}{n+n^2}<\frac{1}{1+n^2}+\frac{2}{2+n^2}+\cdots +\frac{n}{n+n^2}$
$⟹\frac{n(n+1)}{2(n+n^2)}<\frac{1}{1+n^2}+\frac{2}{2+n^2}+\cdots +\frac{n}{n+n^2}$
$\therefore \frac{n(n+1)}{2(n+n^2)}<\frac{1}{1+n^2}+\frac{2}{2+n^2}+\cdots +\frac{n}{n+n^2}<\frac{n(n+1)}{2(1+n^2)}$
Also,
$\underset{n\to \infty }{lim}\frac{n(n+1)}{2(n+n^2)}=\underset{n\to \infty }{lim}\frac{n(n+1)}{2(1+n^2)}=\frac{1}{2}$
Therefore, by Sandwich Theorem:
$\underset{n\to \infty }{lim}\frac{1}{1+n^2}+\frac{2}{2+n^2}+\cdots +\frac{n}{n+n^2}=\frac{1}{2}$