Ratio Test — Question 5

Question 5 – Solution

Step 1: Form and simplify the ratio.

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

Step 2: Use the standard exponential limit.

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

Conclusion.

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