The arithmetic-harmonic mean

Source abstract

Consider two sequences generated by $$a n + 1 = M ( a n , b n ) , b n + 1 = M ′ ( a n + 1 , b n ) , {a_{n + 1}} = M({a_n},{b_n}),\quad {b_{n + 1}} = M’ ({a_{n + 1}},{b_n}),$$ where the a n {a_n} and b n {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 π \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.