- #1
twoflower
- 368
- 0
Hi all,
suppose the following sum:
[tex]
\sum_{n = 1}^{\infty} \frac{n!}{n^{n}}
[/tex]
I had a feeling it should converge, but I can't find a way how to prove that. I tried d'Alembert's criterion:
[tex]
\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} < 1 \Rightarrow \sum a_{n} converges
[/tex]
But the limit is 1 so it doesn't give anything. In fact, if I were able to prove that
[tex]
\lim_{n \rightarrow \infty} \sqrt[n]{n!} = 1
[/tex]
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.
suppose the following sum:
[tex]
\sum_{n = 1}^{\infty} \frac{n!}{n^{n}}
[/tex]
I had a feeling it should converge, but I can't find a way how to prove that. I tried d'Alembert's criterion:
[tex]
\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} < 1 \Rightarrow \sum a_{n} converges
[/tex]
But the limit is 1 so it doesn't give anything. In fact, if I were able to prove that
[tex]
\lim_{n \rightarrow \infty} \sqrt[n]{n!} = 1
[/tex]
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.