On Infinite Process Convergence, Part I: Speeding Series Convergence
|Title||On Infinite Process Convergence, Part I: Speeding Series Convergence|
|Author(s)||Thomas E Phipps|
|Keywords||Infinite Process Convergence, Speeding Series Convergence|
The idea of ?renormalization,? used by physicists for overcoming series divergences by ?subtracting them out,? is generalized here through a reconceptualization of the meaning of discrete infinite process (redefinition of ?value?). The standard mathematical conception, due to Cauchy, approaches ?the infinite? one-sidedly, discarding at the nth stage (n = 1, 2, ?) of a limiting process the whole of whatever may be ?at infinity.? The new concept proposes to treat the process two-sidedly, retaining at each stage of the limiting process an asymptotic approximation to any remainder term ?at infinity.? This accomplishes the same goal as ?summability,? but without modification of finite summands at any stage. Application is made in this paper to speeding convergence of the fast Ramanujan series for pi, and of a still faster-converging series for pi recently discovered. In addition to convergence speeding, the method accomplishes (like summability methods, but in general more powerfully) convergence forcing of ?divergent? processes. As will be shown in later parts, it is applicable not only to series but to continued fractions and all other discrete infinite processes whose ?summands? are known functions of n, required to be asymptotically expandable.