Question

Sum of n!/n^n, n=1 to infinity.
Determine whether the series converges or diverges.

Answer 1

if you start at n=1 (\(0^0\)) is not defined

\[\frac{n!}{n^n}=\frac{(n)(n-1)\cdots(2)(1)}{(n)(n)\cdots(n)(n)}\]

\[=\frac{(n)(n-1)\cdots(3)(2)}{(n)(n)\cdots(n)(n)}\cdot\frac{(2)(1)}{(n)(n)}\le\frac{2}{n^2}\]

Answer 2

a simple comparison test gives the answer

Answer 3

\[0\le \sum_{n=1}^{\infty}\frac{n!}{n^n}\le\sum_{n=1}^{\infty}\frac{2}{n^2}<\infty\]

Answer 4

and again the sum should start at n=1 and not n=0

Answer 5

Yeah I meant for it to say 1. Ill edit that now