# How do you use the Ratio Test on the series sum_(n=1)^oo(n!)/(100^n) ?

May 22, 2018

limnrarroo ((n+1)!)/(100^(n+1))/(n!)/100^n

#### Explanation:

limnrarroo ((n+1)!)/(100^(n+1))/(n!)/100^n

limnrarroo ((n+1)!)/(100^(n+1))*(100^n)/(n!)

$\lim n \rightarrow \infty \frac{\left(n + 1\right)}{{100}^{n + 1}} \cdot \left({100}^{n}\right)$

$\lim n \rightarrow \infty \frac{n + 1}{100}$

$= \frac{\infty}{100} = \infty$

The Ratio Test states that if this limit is greater than 1, the series diverges. Less than 1, it converges. If it is exactly 1, the test is inconclusive.

Because $\infty$ is obviously greater than 1, the series diverges.