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
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
Good job! Indeed the answer is $\displaystyle \frac{1}{e}$
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
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
$\displaystyle \begin{aligned}
\frac{\sqrt[x]{x!}}{x}&=\exp \left[ \ln \left( \prod\limits_{k\,=\,1}^{x}{\left( \frac{k}{x} \right)^{1/x}} \right) \right] \\
& =\exp \left[ \frac{1}{x}\sum\limits_{k\,=\,1}^{x}{\ln \frac{k}{x}} \right] \\
& =\exp \left[ \int_{0}^{1}{\ln y\,dy} \right],\,\text{as }x\to\infty.
\end{aligned}$