Question

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

Answer 1

`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!)`

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

`limnrarroo (n+1)/(100)`

`=oo/100=oo`

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 `oo` is obviously greater than 1, the series diverges.