# How do you find a power series representation for f(x)= 1/(3-x) and what is the radius of convergence?

Oct 11, 2015

The radius of convergence is -3< x <3

#### Explanation:

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

Now compare this epression with the sum of an infinite geometric series 1 +x+x^2 +x^3 +...... . The sum is $\frac{1}{1 - x}$ with -1< x <1.

Now substitute x with $\frac{x}{3}$, then $\frac{1}{1 - \frac{x}{3}} = 1 + \frac{x}{3} + {\left(\frac{x}{3}\right)}^{2} + {\left(\frac{x}{3}\right)}^{3} + \ldots \ldots . .$ where $- 1 < \frac{x}{3} < 1 \to - 3 < x < 3$

Hence $f \left(x\right) = \frac{1}{3 - x} = \frac{1}{3} \left(1 + \left(\frac{x}{3}\right) + {\left(\frac{x}{3}\right)}^{2} + \ldots \ldots . .\right)$

The radius of convergence is therefore -3< x <3