Does this series converge?

Question

I showed that, by a combination of the root test and Stirling's approximation, the series $$\sum \frac{n^n}{n!}$$ converges (the ratio test is inconclusive.) However a solution that I am comparing my work to claims the series diverges.

Who is right?

Thanks,

The series diverges. $n^n\ge n!$ for $n\ge 1$. Did you mean $\sum_{n\ge 1}\frac{n!}{n^n}$? That series converges. — Mark Viola, Dec 24 '15 at 23:54
Hi @dr.mv nope I didn't mean the series you're suggesting. Where's my fault in logic? Using the root test and stirling's approx conveniently left me with $e/\sqrt{2\pi n}$ which goes to zero, and I get convergence by the root test. What do you think? Thanks, — User001, Dec 25 '15 at 00:03
Hi @shailesh the limit is actually e, which is greater than 1. Thanks :-) — User001, Dec 25 '15 at 00:17
... which means the series ..... which is what Brian Scott observed in the first comment very succinctly. So as a learning process, may be you should just put together everything and post it as an answer. I'll definitely upvote that. — Shailesh, Dec 25 '15 at 00:32
Hi @shailesh, unfortunately I am on pen, paper and phone - so I would like to limit my typing during the holiday over the next few days, until I get computer access again. I'll try my best to, though :-) — User001, Dec 25 '15 at 00:50

score 2 · Accepted Answer · answered Dec 25 '15 at 01:16

2

This series does indeed diverge. One way to see this is that, for all $n>1$ we know that $$n^n>n!$$ This is obvious to see because $n^n$ is the product of $n$ terms that are equal to $n$ while $n!$ is the product of $n$ terms that are $\le n$. Because of this, this series is essentially adding infinite terms that are $\ge 1$, which obviously diverges to $\infty$.

answered Dec 25 '15 at 01:16

Will Fisher

5,112

Very nice intuitive answer - earlier I proved this fact using induction, which wasn't hard. But now I see that there's an even easier observation from your approach. Thanks so much @WillFischer :-) – User001 Dec 25 '15 at 01:26

Does this series converge?

1 Answers1