Formal power series

Definition

Let $R$ be a commutative ring. A formal power series in one variable $X$ over $R$ is a sequence $(a_n)_{n \ge 0}$ of elements of $R$, written $\sum_{n \ge 0} a_n X^n$. The set of these series is written $R[\![X]\!]$.

Addition is termwise: $\sum a_n X^n + \sum b_n X^n = \sum (a_n + b_n) X^n$. Multiplication is the Cauchy product: $\left( \sum a_n X^n \right) \left( \sum b_n X^n \right) = \sum_n \left( \sum_{i+j=n} a_i b_j \right) X^n$. With these operations $R[\![X]\!]$ is a commutative ring, and it contains the polynomial ring $R[X]$.

Remarks

The word formal means that no convergence is required. A series is a sequence of coefficients, not a function. A series $\sum a_n X^n$ is a unit in $R[\![X]\!]$ if and only if $a_0$ is a unit in $R$. The coefficient of $X^n$ in a series $f$ is written $[X^n] f$.

Explanations

L1 human

A formal power series is like a polynomial that may go on forever. You add and multiply it like a polynomial, and you never substitute a number for $X$, so the question of convergence does not arise.