How to prove the convergence of a factorial series using d'Alembert's criterion?

[twoflower]

Hi all,

suppose the following sum:

<br /> \sum_{n = 1}^{\infty} \frac{n!}{n^{n}}<br />

I had a feeling it should converge, but I can't find a way how to prove that. I tried d'Alembert's criterion:

<br /> \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} &lt; 1 \Rightarrow \sum a_{n} converges<br />

But the limit is 1 so it doesn't give anything. In fact, if I were able to prove that

<br /> \lim_{n \rightarrow \infty} \sqrt[n]{n!} = 1<br />

I would have it. But I don't know how to make a prove of that...Which criterion or rule should I use here?

Thank you.

 

[twoflower]

In fact, if I were able to prove that

<br /> \lim_{n \rightarrow \infty} \sqrt[n]{n!} = 1<br />

I would have it.

Whoops, the limit isn't 1, it is infinity...

 

[Galileo]

Check d'Alembert again.
\lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_n}| is smaller than 1.

Hint: Make use of: \left(1+\frac{1}{n}\right)^n \to e as n \to \infty

 

[twoflower]

Check d'Alembert again.
\lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_n}| is smaller than 1.

Hint: Make use of: \left(1+\frac{1}{n}\right)^n \to e as n \to \infty

Thank you Galileo, now I can see it goes to 1/e.