What is the series expansion of this polynomial to the negative power? $$
\left(\,
1 + x + {1 \over 2}\,x^{2} + {5 \over 18}\,x^{3} + {25 \over 144}\,x^{4}
\,\right)^{-2}
$$
How would I go about finding the series expansion of this ?.
I know how to use the Taylor expansion, but not how the negative power would affect the answer. 

The Taylor series for $f$ about $x=0$ is
$$
\sum_{n=0}^\infty c_nx^n \quad\text{ where }\quad c_n={f^{(n)}(0)\over n!},
$$
which in this case for your $f(x)$ above yields 
$$
1-2x+2x^2-{14\over 9}x^3+{77\over 72}x^4-{11x^5\over 24}+\cdots
$$
