In the many years since Archimedes' discovery, algorithms of a similar type have come up repeatedly. Perhaps no one understood them so well as Carl Friedrich Gauss (1777--1855); Gauss considered the following iteration scheme:

Start with and . For , define

[This same algorithm is sometimes called *Borchardt's algorithm*.
It is equivalent to Archimedes' algorithm; just
substitute and
.]

Gauss was considering a difficult problem when he encountered this sequence. He asked his teacher, Pfaff, about the sequence; Pfaff showed that for any positive and , the common limit of the two sequences is

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Neal Carothers - carother@bgnet.bgsu.edu