Taylor series coefficients of general rational function

Question

Given two (possibly infinite) sequences of complex numbers $\{a_1, a_2, \ldots\}$ and $\{b_1, b_2, \ldots\}$, I'm looking to find a closed form expression for each element of the sequence of complex numbers $\{c_1, c_2, \ldots\}$ where the sequences are related by

$$\frac{1+\sum_{n=1}^\infty a_n x^n}{1+\sum_{m=1}^\infty b_m x^m} = 1+\sum_{r=0}^\infty c_r x^r\,.$$

Essentially, I am looking for a rapid computational algorithm to find the Taylor series expansion coefficients of a general rational function.

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Edit: The problem can be simplified to finding a series expansion of $(1+\sum_{m=1}^\infty b_m x^m)^{-1}$ since then the $c$ coefficients are given by the series product.

Answer 1

Hint: The coefficients $c_r$ in terms of determinants of the coefficients of $1+\sum_{m=1}^\infty b_m x^m$ can be found here.

Answer 2

If you actually want to compute the coefficients, here is the standard way:

Dividing an infinite power series by another infinite power series