# What is the sum of the infinite geometric series 2+2/3+2/9+2/27+...?

Jun 22, 2016

$3$

#### Explanation:

a geometric series can be defined as: ${a}_{n} = {a}_{1} \cdot {\left(r\right)}^{n - 1}$ where ${a}_{1}$ is the first value of the series and $r$ is the common ratio.

The common ratio is $\frac{1}{3}$ and the first term is $2$ so our series is:

${a}_{n} = 2 \cdot {\left(\frac{1}{3}\right)}^{n - 1}$

You can only sum an infinite geometric series if it is <b>convergent</b>, that is, that it converges to one value. A series is convergent if $\left\mid r \right\mid < 1$ which in this case it is.

The equation to sum an infinite series is: ${a}_{1} / \left(1 - r\right)$ so by plugging all values we get: $\frac{2}{1 - \frac{1}{3}}$ which is $\frac{2}{\frac{2}{3}}$ or $\frac{6}{2}$ which simplifies to $3$