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The Arithmetic-Geometric Mean of Gauss

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Pi: A Source Book

Abstract

The arithmetic-geometric mean of two numbers a and b is defined to be the common limit of the two sequences \(\left\{ {{a_n}} \right\}_{n = 0}^\infty \) and \(\left\{ {{b_n}} \right\}_{n = 0}^\infty \) determined by the algorithm

$${a_0} = a,\,{b_0} = b, $$
$${a_{n + 1}} = \left( {{a_n} + {b_n}} \right)/2,\,{b_{n + 1}} = {\left( {{a_n}{b_n}} \right)^{1/2}},\,n = 0,1,2, \ldots \,.$$
((0.1))

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  1. Department of Mathematics, Amherst College, Amherst, MA, 01002, USA

    by David A. Cox

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Cox, D.A. (2000). The Arithmetic-Geometric Mean of Gauss. In: Pi: A Source Book. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3240-5_55

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  • DOI: https://doi.org/10.1007/978-1-4757-3240-5_55

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4757-3242-9

  • Online ISBN: 978-1-4757-3240-5

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