# How do you apply the ratio test to determine if sum_(n=1)^oo 3^n is convergent to divergent?

Jan 15, 2018

The series diverges.

#### Explanation:

The the $n$th term in the series is given by ${a}_{n} = {3}^{n} .$

The ratio test states that the series should be convergent if:

${\lim}_{n \to \infty} {a}^{n + 1} / {a}^{n} < 1$

So, in our case we have:

${\lim}_{n \to \infty} {3}^{n + 1} / {3}^{n} = {\lim}_{n \to \infty} {3}^{n + 1 - n} =$

$= {\lim}_{n \to \infty} 3 = 3$

$3 > 1$ so the ratio test tells us this series diverges.