<s>[INST] 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!
 [/INST] Hint: Start with
$$
\frac {4x} {x^2-2x-3} = \frac {4x}{(x-1)^2-4}
=\frac {x-3+3(x+1)}{(x-3)(x+1)}.
$$
details:
then you get to
$$
 \frac {4x} {x^2-2x-3} = \frac 1{x+1} + \frac 3{x-3}
= \sum_{n=0}^\infty (-1)^nx^n + \sum_{n=0}^\infty -3^{-n}x^n.
$$
</s>