The arithmetic-harmonic mean

Abstract:

Consider two sequences generated by \[ {a_{n + 1}} = M({a_n},{b_n}),\quad {b_{n + 1}} = M’ ({a_{n + 1}},{b_n}),\] where the ${a_n}$ and ${b_n}$ 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 $\pi$. 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.