_ n (1+ 1+1 2 + .+1 n n^ 2 )^ n =......... - Vidyadip

MCQ

$\lim _{n \rightarrow \infty}\left(1+\frac{1+\frac{1}{2}+\ldots \ldots .+\frac{1}{n}}{n^{2}}\right)^{n}=.........$

A
$\frac{1}{2}$
B
$0$
C
$\frac{1}{ e }$
✓
$1$

✓

Answer

Correct option: D.

$1$

d
Given limit is of $1^{\infty}$ form

$\text { So, } l=\exp \left(\lim _{n \rightarrow \infty} \frac{1+\frac{1}{2}+\frac{1}{3}+\ldots \ldots .+\frac{1}{n}}{n}\right)$

Now,

$0 \leq 1+\frac{1}{2}+\frac{1}{3}+\ldots .+\frac{1}{n} \leq 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n}}$

$\leq 2 \sqrt{n}-1$

So, $l=\exp (0)$ (from sandwich theorem)

$=1$

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