Ratio Test — Question 5

*Question 5 Use the Ratio Test, without Stirling’s formula, to determine whether $\displaystyle\sum_{n=1}^{\infty}\frac{n!}{n^n}$ converges or diverges. Show all algebra used to find the ratio limit.

*Question 5 – Solution

$$\begin{align*} \frac{a_{n+1}}{a_n}&=\frac{(n+1)!}{(n+1)^{n+1}}\frac{n^n}{n!}=\frac{(n+1)n^n}{(n+1)^{n+1}}\\&=\left(\frac{n}{n+1}\right)^n=\left(1+\frac1n\right)^{-n}. \end{align*}$$

Since $(1+1/n)^n\to e$, $$L=\lim_{n\to\infty}\left(1+\frac1n\right)^{-n}=e^{-1}<1.$$

The series converges. This gives the decisive geometric-rate factor $1/e$ directly, without an asymptotic formula for $n!$.