---
tag: 0H6KWAYS87
kind: statement
role: theorem
name: Product of power series
msc: [13F25]
uses: [072W67KFYK, 0M98XYVX6S]
sources:
  - url: https://en.wikipedia.org/wiki/Cauchy_product
    locator: Cauchy product of two power series
---

## Statement

Let $R$ be a commutative ring, let $f, g \in R[\![X]\!]$ be [[formal power series|072W67KFYK]], and let $n$ be a natural number.
Then
$$[X^n](f g) = \sum_{i + j = n} [X^i] f \cdot [X^j] g = \sum_{i=0}^{n} [X^i] f \cdot [X^{n-i}] g.$$

## Remarks

In many texts this formula is the definition of the product of formal power series.
The sequence of coefficients of $fg$ is the convolution of the sequences of $f$ and $g$. This is the Cauchy product.
Example: $[X^n] \frac{1}{(1 - X)^2} = \sum_{i=0}^{n} 1 \cdot 1 = n + 1$.

## Explanations

### L1 human

To find the coefficient of $X^n$ in a product, collect all ways to get $X^n$ as $X^i$ times $X^j$.
Add the products of the matching coefficients.