The arithmetic-harmonic mean

Consider two sequences generated by an1+ I = M(a,,. b,), h,1 + =M'( a, I, b,) where the a,, and b,, are positive and M and M' are means. The paper discusses the nine processes which arise by restricting the choice of M and M' to the arithmetic, geometric and harmonic means, one case being that used by Archimedes to estimate 1r. Most of the paper is devoted to the arithmetic-harmonic mean, whose limit is expressed as an infinite product and as an infinite series in two ways. 1. Introduction. Recently (3) we have discussed the generalized Archimedean process in which two sequences (an) and (bn) are defined by (la) an+ 1 = M(an, bn), (lb) bn+1 = M'(an+11 bn), where a0, bo E R+ and M and M' are mappings from R+ x R+ to R+ which satisfy