Math Help - interesting limit

1. interesting limit

For all of the following prove without using l'Hopital's rule.

i) Does $\displaystyle\lim_{n\to\infty}\left(1+\frac{1}{\sq rt{n}}\right)^n$ converge? If so what is its limit? If not prove that it diverges.

ii) For what values $p\in\mathbb{R}$ does $\displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n^p }\right)^n$ converge? If possible give a formula for the converging value.

2. Originally Posted by putnam120
For all of the following prove without using l'Hopital's rule.

i) Does $\displaystyle\lim_{n\to\infty}\left(1+\frac{1}{\sq rt{n}}\right)^n$ converge? If so what is its limit? If not prove that it diverges.

ii) For what values $p\in\mathbb{R}$ does $\displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n^p }\right)^n$ converge? If possible give a formula for the converging value.
Maybe this will help.

3. thanks. I solved them by using the binomial theorem but that way works also. thanks once more.

4. Originally Posted by putnam120
For all of the following prove without using l'Hopital's rule.

i) Does $\displaystyle\lim_{n\to\infty}\left(1+\frac{1}{\sq rt{n}}\right)^n$ converge? If so what is its limit? If not prove that it diverges.

ii) For what values $p\in\mathbb{R}$ does $\displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n^p }\right)^n$ converge? If possible give a formula for the converging value.
You need to know that:

$
\lim_{x \to \infty}\left(1+\frac{1}{x}\right)^x = e
$

Then for $x$ large enough:

$
\left(1+\frac{1}{x}\right)^x=e+b_n
$

and $|b_n|<1$.

So:

$
(e-1)^{\sqrt{n}}<\left(1+\frac{1}{\sqrt{n}}\right)^n< (e+1)^{\sqrt{n}}
$

and the rest of part (i)should follow from the squeeze theorem.

RonL