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An approximation to the arithmetic-geometric mean

Published online by Cambridge University Press:  23 January 2015

G. J. O. Jameson
G. J. O. Jameson*
Affiliation:
Dept. of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF e-mail: g.jameson@lancaster.ac.uk
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Extract

Given positive numbers a > b, consider the ‘agm iteration’ given by a 0 = a, b 0 = b and

At each stage, the two new numbers are the arithmetic and geometric means of the previous two. It is easily seen that bn < an, (an) is decreasing, (bn ) is increasing and an + 1 − bn + 1 < ½(an − bn), and hence that (an ) and (bn ) converge to a common limit, which is called the arithmetic-geometric mean of a and b. We will denote it by M (a, b).

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Articles
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The Mathematical Gazette , Volume 98 , Issue 541 , March 2014, pp. 85 - 95
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Copyright © The Mathematical Association 2014

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