What rational function does this power series represent

Question

I cannot find a good explanation on how to represent a power series as a rational function. Thanks!

$$2 - \frac{2}{3}x + \frac{2}{9}x^2 - \frac{2}{27}x^3 + ... $$

Answer 1

To spell this out, your series can be written in the form of a geometric series:

\begin{eqnarray}2\left(1-\left(\dfrac{x}{3}\right)+\left(\dfrac{x}{3}\right)^{2}-\cdots\right)\end{eqnarray}

You can think of this as an infinite geometric series with $r=-\frac{x}{3}$ and $a=2$, so the fractional form is:

\begin{eqnarray}2\frac{1}{1+\frac{x}{3}} &=& \frac{6}{x+3}\end{eqnarray}

Answer 2

$2\left(1-\left(\dfrac{x}{3}\right)+\left(\dfrac{x}{3}\right)^{2}-\cdots\right)$, geometric.

Answer 3

The Taylor series expansion of $\frac{6}{x+3}$ at $x=0$

$$\frac{6}{x+3} = 2 - \frac{2 x}{3} + \frac{2 x^2}{9} - \frac{2 x^3}{27} + \frac{2 x^4}{81} + O(x^5)$$