Math 2.0

Commutative algebra Definition

Coefficient of a power series

Builds on Formal power series

Definition

Let RR be a commutative ring and let nn be a natural number. The coefficient of XnX^n is the map [Xn]:R[ ⁣[X] ⁣]→R[X^n] : R[\![X]\!] \to R that sends a formal power series f=∑m≥0amXmf = \sum_{m \ge 0} a_m X^m to ana_n.

Remarks

The map [Xn][X^n] is RR-linear: [Xn](f+g)=[Xn]f+[Xn]g[X^n](f + g) = [X^n] f + [X^n] g and [Xn](rf)=r [Xn]f[X^n](r f) = r\, [X^n] f. Two series are equal if and only if all their coefficients are equal. The coefficient of a product is given by the node product of power series.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Formal_power_series, Extracting coefficients
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