Ratio Test — Question 7

*Question 7 Determine whether $\displaystyle\sum_{n=1}^{\infty}\frac{n!}{n^{10}}$ converges or diverges.

1. Simplify $a_{n+1}/a_n$.

2. Use its limiting behavior to justify the conclusion.

3. Explain why the nth-term test also applies.

*Question 7 – Solution

$$\begin{align*} \frac{a_{n+1}}{a_n}&=\frac{(n+1)!}{(n+1)^{10}}\frac{n^{10}}{n!}=(n+1)\left(\frac{n}{n+1}\right)^{10}\\&=n\left(1+\frac1n\right)^{-9}. \end{align*}$$

The factor $(1+1/n)^{-9}\to1$, while $n\to\infty$. Hence $L=\infty>1$.

The series diverges. Its terms eventually grow rather than approach zero, so the nth-term divergence test confirms the result. A factorial eventually dominates every fixed power $n^{10}$.