Find Limit: Evaluate \frac{n}{(n!)^{\frac1n}}

[Ali 2]

Hi ,

Evaluate the following limit :

\lim_{n\rightarrow\infty}\frac{n}{(n!)^{\frac1n}}=\lim_{n\rightarrow\infty}\frac{n}{\sqrt[n]{n!}}

 

[NateTG]

Do you know about stirling's approximation?
http://mathworld.wolfram.com/StirlingsApproximation.html

n! \approx \sqrt{(2n+\frac{1}{3})\pi} n^n e^{-n}

Which readily leads to the answer that the limit is e

 

[Ali 2]

I know about that .. the solution will be obtained easily by that method ..



but.. could you solve the question without stirling's approximation ?

 

[dextercioby]

I know about that .. the solution will be obtained easily by that method...but...could you solve the question without stirling's approximation ?

I'm afraid Stirling's approximation provides the simplest approach.Keep in mind that u've to compute the limit of a sequence and u cannot make the transition to a function,due to a factorial in the denominator.Of course,that factorial can be put under the form
n!=\Gamma(n+1)
,but that won't do you any good,since it still involves discrete values for "n".

Daniel.