The arithmetic-geometric mean,denoted , of two non-negative numbers is defined as the limit of two sequences , with

where

This sequence converges very quickly.

## Properties

• Because the arithmetic mean approaches from above, and the geometric mean approaches from below, the two can be used to estimate an error value during calculation. For correct decimal places, it must be the case that for the th iteration of calculation

## Proof of Existence

This yields that

is a non-decreasing sequence.

Because the mean of two numbers lies between the two numbers, there exists a lower bound for . Then by the Monotone Convergence Theorem, there exists a limit such that

Using the definition of the sequence

This implies that

Taking the limit as , and using the Algebra of limits

This completes the proof.