---
tag: PCYRWA7SYB
kind: concept
name: Convergence of an infinite product
msc: [40A20]
sources:
  - url: https://en.wikipedia.org/wiki/Infinite_product
---

## Summary

An [[infinite product|JYV1Y8XX16]] converges when, after finitely many zero factors are set aside, its partial products tend to a nonzero limit.
The nonzero condition makes the theory parallel to series: a product of positive terms converges exactly when the series of logarithms converges.
A convergent product has factors tending to $1$, and its value is $0$ only when some factor is $0$.
The exact definition is in [[convergence of an infinite product|H1Z2TKY3EQ]].

## Explanations

### L1 human

An infinite product converges when multiplying more and more factors brings you closer and closer to a fixed nonzero number.
Products whose partial products shrink to $0$ are said to diverge to $0$, just as a sum can diverge to infinity.
For example, $\tfrac12 \cdot \tfrac23 \cdot \tfrac34 \cdots$ has partial products $1/(N+1)$, so it diverges to $0$.