Question #13952

Dec 7, 2017

Diverges.

Explanation:

Rearranging a little:

$\left(\frac{1}{3}\right) {\sum}_{k = 0}^{\infty} \left({5}^{k}\right)$

This is a geometric series with $r = 5$. Since $r > 1$ we know that the series diverges.

Dec 7, 2017

Divergent.

Explanation:

This is a geometric series with common ratio of $5$.

The sum of a geometric series is given by:

$a \left(\frac{1 - {r}^{n}}{1 - r}\right)$

Where $a$ is the first term, $r$ is the common ratio and $n$ is the nth term.

If $| r | < 1$

Then the sum to infinity is:

$\frac{a}{1 - r}$

In this example:

The common ratio $r$ is $5$:

$| 5 | > 1$

So the series diverges.