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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.

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.


marked as duplicate by Community Mar 14 '18 at 22:16

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.


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.

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    $\begingroup$ That is a very pretty formula! $\endgroup$ – Jair Taylor Mar 14 '18 at 17:54
  • $\begingroup$ It may take quite a while. $\endgroup$ – marty cohen Mar 14 '18 at 21:50
  • $\begingroup$ Thanks; my question is a duplicate of the one linked to. Flagging my question as duplicate. $\endgroup$ – QuantumDot Mar 14 '18 at 21:53
  • $\begingroup$ @QuantumDot: You're welcome. $\endgroup$ – Markus Scheuer Mar 14 '18 at 21:55

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

Dividing an infinite power series by another infinite power series


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