Ratio Test — Question 7

Question 7 – Solution

Step 1: Compute the ratio.

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

Step 2: Evaluate its behavior.

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

Conclusion.

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 $n^{10}$.