A linear transformation to accelerate the convergence of the negative binomial series

TL;DR

This paper introduces a linear transformation that speeds up the convergence of the negative binomial series, enabling more efficient computation of functions like ln(1+x) and the incomplete beta function.

Contribution

A novel linear sequence transformation is proposed to accelerate the negative binomial series convergence, extending its applicability.

Findings

Accelerates convergence of negative binomial series

Enables extended use of Taylor expansion for ln(1+x)

Improves computation of incomplete beta function

Abstract

A linear sequence transformation is defined that accelerates the convergence of the negative binomial series when the terms of the binomial have the same sign. The transformed series can be used to extend the region of applicability of the Taylor expansion of ln(1+x) and to compute the incomplete beta function.