interesting limit

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August 29th 2007, 06:38 PM
putnam120

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.
August 29th 2007, 06:59 PM
ThePerfectHacker

Quote:

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.
August 29th 2007, 08:42 PM
putnam120

thanks. I solved them by using the binomial theorem but that way works also. thanks once more.
August 29th 2007, 08:58 PM
CaptainBlack

Quote:

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