mean can be calculated by an analogous method, using sequences of geometric and harmonic means. One finds that . The arithmetic–harmonic mean can be similarly defined, but takes the same value as the geometric mean (see section "Calculation" there). The arithmetic–geometric mean can be used to compute – among others – logarithms, complete and incomplete elliptic integrals of the first and second kind, and Jacobi elliptic functions. Proof of existence From the inequality of arithmetic and geometric means we can conclude that: and thus that is, the sequence is nondecreasing. Furthermore, it is easy to see that it is also bounded above by the larger of and (which follows from the fact that both the arithmetic and geometric means of two numbers lie between them). Thus, by the monotone convergence theorem, the sequence is convergent, so there exists a such that: However, we can also see that: and so: Q.E.D. Proof of the integral-form expression This proof is given by Gauss. Let Changing the variable of integration to , where gives Thus, we have The last equality comes from observing that . Finally, we obtain the desired result Applications The number π For example, according to the Gauss–Legendre algorithm: where with and , which can be computed without loss of precision using Complete elliptic integral K(sinα) Taking and yields the AGM where is a complete elliptic integral of the first kind: That is to say that this quarter period may be efficiently computed through the AGM, Other applications Using this property of the AGM along with the ascending transformations of John Landen, Richard P. Brent suggested the first AGM algorithms for the fast evaluation of elementary transcendental functions (, , ). Subsequently, many authors went on to study the use of the AGM algorithms. See also Landen's transformation Gauss–Legendre algorithm Generalized mean References Notes Citations Sources Means Special functions Elliptic functions Articles containing proofs