---
tag: JYV1Y8XX16
kind: concept
name: Infinite product
msc: [40A20]
sources:
  - url: https://en.wikipedia.org/wiki/Infinite_product
same_as:
  wikidata: Q2266843
  wikipedia: Infinite product
---

## Summary

An infinite product $\prod_{n=1}^{\infty} a_n$ is the multiplicative analogue of a [[series|0NM626FH6Q]]: it is the sequence of partial products $a_1 a_2 \cdots a_N$.
Its value, when it [[converges|PCYRWA7SYB]], is the limit of these partial products.
Infinite products express many classical functions and constants, such as the [[Wallis product|0XCYA4YERS]] for $\pi$ and the [[Euler product formula|0C3PN4075R]] for the zeta function.
The exact definition is in [[infinite product|RZ0Z2BTMT4]].

## Explanations

### L1 human

An infinite product multiplies infinitely many numbers, one after another.
As with infinite sums, one multiplies the first $N$ factors and looks at what happens as $N$ grows.
For the product to settle down to a nonzero number, the factors must get closer and closer to $1$.
For example, $(1 - \tfrac14)(1 - \tfrac19)(1 - \tfrac1{16}) \cdots$ has partial product $(N+1)/(2N)$ up to the factor $1 - 1/N^2$, and these approach $\tfrac12$.