Acceleration of Series

Abstract

The rate of convergence of infinite series can be accelerated b y a suitable splitting of each term into two parts and then combining the second part of the n-th term with the first part of the (n+1) -th term t get a new series and leaving the first part of the first term as an "orphan". Repeating this process an infinite number of times, the series will often approach zero, and we obtain the series of orphans, which may converge faster than the original series. H euristics for determining the splits are given. Various mathematical constants, originally defined as series having a term ratio which approaches 1, are accelerated into series having a term ratio less than 1. This is done with the constants of Euler and Catalan. The se ries for pi/4 = arctan 1 is transformed into a variety of series, among which is one having a term ration of 1/27 and another having a term ratio of 54/3125. A series for 1/pi is found which has a term ratio of 1/64 and each term of which is an integer divided by a powe r of 2, thus making it easy to evaluate the sum in binary arithmetic. We express zeta(3) in terms of pi-3 and a series having a term ra tio of 1/16. Various hypergeometric function identities are found, as well as a series for (arcsin y)-2 curiously related to a series f or y arcsin y. Convergence can also be accelerated for finite sums, as is shown for the harmonic numbers. The sum of the reciprocals of the Fibonacci numbers has been expressed as a series having the convergence rate of theta function. Finally, it is shown that a series whose n-th term ratio is (n+p)(n+q)/(n+r)(n+s), where p, q, r, s are integers, is equal to c + d pi-2, where c and d are rational.