# Power series?

## Be sure to indicate the interval on which the power series converges. Thanks!

May 16, 2018

Recall the sum of a geometric series is given by

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

We can rewrite our sequence as

$g \left(x\right) = \frac{2}{1 - \frac{1}{3} x}$

$g \left(x\right) = \frac{\frac{2}{3}}{1 - \frac{1}{3} x}$

Now we can match up coefficients to get

$a = \frac{2}{3}$ and $r = \frac{1}{3}$

Thus the power series is

$g \left(x\right) = \frac{2}{3} {\left(\frac{1}{3} x\right)}^{n}$

The interval of convergence will be $\left(- 3 , 3\right)$ (because whenever we stray out of this interval we can immediately see that the geometric series diverges.

Hopefully this helps!