# limit of xth root etc etc

• Apr 25th 2008, 02:45 PM
polymerase
limit of xth root etc etc
Determine if $\displaystyle \lim_{x\to\infty} \frac{\sqrt[x]{x!}}{x}$ exist (and value) and if it doesn't explain why.
(in case thats too small, it reads the xth root of x! all over x)

Potentially VERY easy series questions?!?

Thanks
• Apr 25th 2008, 02:50 PM
Mathstud28
Quote:

Originally Posted by polymerase
Determine if $\displaystyle \lim_{x\to\infty} \frac{\sqrt[x]{x!}}{x}$ exist (and value) and if it doesn't explain why.
(in case thats too small, it reads the xth root of x! all over x)

Potentially VERY easy series questions?!?

Thanks

use the fact taht $\displaystyle n!\sim{\sqrt{2\pi{n}}n^{n}e^{-n}}$

then it should be apparent
• Apr 25th 2008, 03:01 PM
polymerase
Quote:

Originally Posted by Mathstud28
use the fact taht $\displaystyle n!\sim{\sqrt{2\pi{n}}n^{n}e^{-n}}$

then it should be apparent

Firstly, thank you

Secondly, so is the answer $\displaystyle \frac{1}{e}$?

Thirdly, is there another way to do this, because we never learned the relationship between n! and the stuff you stated above :)
• Apr 25th 2008, 03:06 PM
Mathstud28
Quote:

Originally Posted by polymerase
Firstly, thank you

Secondly, so is the answer $\displaystyle \frac{1}{e}$?

Thirdly, is there another way to do this, because we never learned the relationship between n! and the stuff you stated above :)

Good job! Indeed the answer is $\displaystyle \frac{1}{e}$ (Clapping)

and you could also do this $\displaystyle y=\lim_{x\to\infty}\frac{(x!)^{\frac{1}{x}}}{x}\Ri ghtarrow{\ln(y)=\frac{\ln(x!)}{x^2}}$

and this is painful but $\displaystyle \ln(x!)\approx{x\ln(x)-x+\frac{\ln(x)}{2}+\frac{\ln(2\pi)}{2}}$

Or you could als use that $\displaystyle n!=\int_0^{\infty}x^{n}e^{-x}dx$

The gamma function and go L'hopitals
• Apr 25th 2008, 03:13 PM
polymerase
Quote:

Originally Posted by Mathstud28
Good job! Indeed the answer is $\displaystyle \frac{1}{e}$ (Clapping)

and you could also do this $\displaystyle y=\lim_{x\to\infty}\frac{(x!)^{\frac{1}{x}}}{x}\Ri ghtarrow{\ln(y)=\frac{\ln(x!)}{x^2}}$

and this is painful but $\displaystyle \ln(x!)\approx{x\ln(x)-x+\frac{\ln(x)}{2}+\frac{\ln(2\pi)}{2}}$

Or you could als use that $\displaystyle n!=\int_0^{\infty}x^{n}e^{-x}dx$

The gamma function and go L'hopitals

REALLY?! thats the alternative! how the %$!@ does our prof. expect us to solve that? we never learn ANY of that! LOL is there ANOTHER way?:D • Apr 25th 2008, 03:13 PM Mathstud28 Quote: Originally Posted by Mathstud28 Good job! Indeed the answer is$\displaystyle \frac{1}{e}$(Clapping) and you could also do this$\displaystyle y=\lim_{x\to\infty}\frac{(x!)^{\frac{1}{x}}}{x}\Ri ghtarrow{\ln(y)=\frac{\ln(x!)}{x^2}}$and this is painful but$\displaystyle \ln(x!)\approx{x\ln(x)-x+\frac{\ln(x)}{2}+\frac{\ln(2\pi)}{2}}$Or you could als use that$\displaystyle n!=\int_0^{\infty}x^{n}e^{-x}dx$The gamma function and go L'hopitals Expanding upon my first thought after simplifying you would have$\displaystyle \lim_{n\to\infty}\frac{\ln(x)}{x}-1+\frac{\ln(x)}{2x^2}+\frac{\ln(2\pi)}{2x^2}=-1$therefore$\displaystyle \ln(y)=-1\Rightarrow{y=e^{-1}}$This came from the ln(x!) approximation over x² • Apr 25th 2008, 03:15 PM Mathstud28 Quote: Originally Posted by polymerase REALLY?! thats the alternative! how the %$!@ does our prof. expect us to solve that? we never learn ANY of that!

LOL is there ANOTHER way?:D

Haha...Uhm..I am surprised I thought of three ways...uhm...I cant think of anymore...maybe if you look at this post above you cna use that...that is not overly complicated? haha
• Apr 25th 2008, 03:23 PM
polymerase
Quote:

Originally Posted by Mathstud28
Expanding upon my first thought after simplifying you would have $\displaystyle \lim_{n\to\infty}\frac{\ln(x)}{x}-\frac{1}{x}+\frac{\ln(x)}{2x^2}+\frac{\ln(2\pi)}{2 x^2}=-1$ therefore $\displaystyle \ln(y)=-1\Rightarrow{y=e^{-1}}$

This came from the ln(x!) approximation over x²

How do you figure out/calculate that $\displaystyle \ln (x!)=$above stuff?
• Apr 25th 2008, 03:24 PM
Mathstud28
Quote:

Originally Posted by polymerase
How do you figure out/calculate that $\displaystyle \ln (x!)=$above stuff?

Uhm....well I have the memorized but if you need to see the entire thing...hold on let me find...an ariticle...ah! here we go read here

Natural logarithm - Wikipedia, the free encyclopedia

sorry edit:
Look here
Factorial - Wikipedia, the free encyclopedia
• Apr 25th 2008, 08:32 PM
Isomorphism
Quote:

Originally Posted by Mathstud28
Or you could als use that $\displaystyle n!=\int_0^{\infty}x^{n}e^{-x}dx$

How will you use this? (Wondering)
• May 18th 2008, 06:50 AM
Krizalid
Quote:

Originally Posted by polymerase

LOL is there ANOTHER way?:D

\displaystyle \begin{aligned} \frac{\sqrt[x]{x!}}{x}&=\exp \left[ \ln \left( \prod\limits_{k\,=\,1}^{x}{\left( \frac{k}{x} \right)^{1/x}} \right) \right] \\ & =\exp \left[ \frac{1}{x}\sum\limits_{k\,=\,1}^{x}{\ln \frac{k}{x}} \right] \\ & =\exp \left[ \int_{0}^{1}{\ln y\,dy} \right],\,\text{as }x\to\infty. \end{aligned}
• May 18th 2008, 05:25 PM
arbolis
Nice limit question. Using my intuition I thought it would tend to positive infinite.