# Is the series \sum_(n=0)^\infty1/((2n+1)!) absolutely convergent, conditionally convergent or divergent?

## use the appropriate test... I know Root wouldn't work with a factorial, but I got stuck on Ratio, too. I am at $\setminus \stackrel{L}{\setminus \infty} | \frac{1}{2 n + 2} |$, what should I put next?

Apr 23, 2018

"Compare it with "sum_{n=0}^oo 1/(n!) = exp(1) = e = 2.7182818...

#### Explanation:

$\text{Each term is equal to or smaller than the}$

sum_{n=0}^oo 1/(n!) = exp(1) = e = 2.7182818...

$\text{All terms are positive so the sum S of the series is between}$

$0 < S < e = 2.7182818 \ldots .$

$\text{So the series is absolutely convergent.}$