[Math] Finding the first non-zero terms of a power series

Question

I have the function:
$f(x) = \frac{30}{(x^2 + 1)(x^2-9)}$

I need to find the first four non-zero terms of the power series centered at zero. I have not had much experience with power series so I am not sure how to start/complete this problem.

Answer 1

$f(x)=\frac{-3}{x^2+1}+\frac{3}{x^2-9}$

$\frac{-3}{x^2+1}=-3(1-x^2+x^4-x^6+....)$

$\frac{3}{x^2-9}=\frac{-1/3}{1-x^2/9}=(-1/3)(1+x^2/9+x^4/81+x^6/729+.....)$

Combine to get $f(x)=-4/3+(80/27)x^2-........$

I'll let you finish.