# Use divergent test to determine convergence of the following series?

## ${\sum}_{n = 0}^{\infty} \left({3}^{n + 1} / {e}^{n}\right)$

Mar 25, 2018

Diverges.

#### Explanation:

The divergence test, when applied to a series $\sum {a}_{n}$ simply entails taking ${\lim}_{n \to \infty} {a}_{n}$:

If ${\lim}_{n \to \infty} {a}_{n} \ne 0 ,$ the series $\sum {a}_{n}$ diverges.

Note, if ${\lim}_{n \to \infty} {a}_{n} = 0 ,$ the series doesn't necessarily converge. It may, but it's not guaranteed.

So,

${\lim}_{n \to \infty} {3}^{n + 1} / {e}^{n} = {\lim}_{n \to \infty} \frac{{3}^{n} {3}^{1}}{e} ^ n = 3 {\lim}_{n \to \infty} {\left(\frac{3}{e}\right)}^{n} = 3 \left(\infty\right) = \infty$

Thus, the series diverges.