# Finding Power Series Representations

For f(x), find a power series representation centered at the given value of a and determine the radius of convergence.

$$f(x) = \frac {4x} {x^2-2x-3} ; a=0.$$

How would i begin this?
And what does it mean at the given value of a?
Thanks!

• Do you know what it means to expand a function into a power series around a point? (It gives you the point; $x=0$...)
– blue
Mar 27, 2014 at 22:23
• I just know about a function to power series, but not at a given point Mar 27, 2014 at 22:25
• "At a point $a$" just means the series should be of the form $\sum a_n(x-a)^n$.
– OR.
Mar 27, 2014 at 22:25
• Does that mean we sub the value in for a? Mar 27, 2014 at 22:30

Hint: Start with $$\frac {4x} {x^2-2x-3} = \frac {4x}{(x-1)^2-4} =\frac {x-3+3(x+1)}{(x-3)(x+1)}.$$
then you get to $$\frac {4x} {x^2-2x-3} = \frac 1{x+1} + \frac 3{x-3} = \sum_{n=0}^\infty (-1)^nx^n + \sum_{n=0}^\infty -3^{-n}x^n.$$