Coefficient of a power series

Definition

Let $R$ be a commutative ring and let $n$ be a natural number. The coefficient of $X^n$ is the map $[X^n] : R[\![X]\!] \to R$ that sends a formal power series $f = \sum_{m \ge 0} a_m X^m$ to $a_n$.

Remarks

The map $[X^n]$ is $R$-linear: $[X^n](f + g) = [X^n] f + [X^n] g$ and $[X^n](r f) = r\, [X^n] f$. 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.

Explanations

L1 human

The coefficient $[X^n] f$ is the number in front of $X^n$ in the series $f$. For $f = 1 + 2X + 3X^2 + \dots$, the coefficient $[X^2] f$ is $3$.