---
tag: 072W67KFYK
kind: definition
name: Formal power series
msc: [13F25]
defines: 033WGN00BF
sources:
  - url: https://en.wikipedia.org/wiki/Formal_power_series
    locator: Definition of the formal power series ring
---

## 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|0M98XYVX6S]] 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.