Power Series Expansion Using Partial Fractions

The partial-fraction decomposition of a rational function is not only an algebraic simplification -- it is also the fastest route to its power-series expansion. Once a proper fraction is written in partial fractions, each term of the form $\dfrac A{1-kx}$ is immediately recognisable as a geometric series,

and a repeated-factor term $\dfrac A{(1-kx)^2}$ expands using the standard derivative-of-a-geometric-series identity $\dfrac1{(1-kx)^2}=\displaystyle\sum_{r=0}^\infty (r+1)(kx)^r$. Summing all the partial-fraction terms' expansions together, term by term, gives the power series of the original rational function, and the coefficient of any chosen power (of $x^n$ in general, or of a specific power such as $x^3$ or $x^4$) can be read straight off this sum as a closed formula.

A factor written as $a-x$ rather than $1-kx$ is first rearranged, $\dfrac A{a-x}=\dfrac Aa\cdot\dfrac1{1-x/a}$, so it fits the same standard geometric-series form with $k=1/a$. …