On Taylor series expansion of $(1+ z)^{A}$ for $|z|>1$

TL;DR

This paper develops a method to express the function $(1+ z)^{A}$ as a product of convergent series for $|z|>1$, where traditional Taylor series do not converge, expanding the understanding of such functions.

Contribution

The paper introduces a novel series expansion for $(1+ z)^{A}$ valid for $|z|>1$, overcoming the limitations of standard Taylor series.

Findings

Derived a convergent series representation for $(1+ z)^{A}$ when $|z|>1$

Provided a practical method for series expansion in the complex plane

Extended the applicability of series expansions beyond the radius of convergence

Abstract

It is well known that the Taylor series expansion of $(1+ z)^{A}$ does not converge for $|z|>1$ where A is a real number which is not equal to zero or a positive integer. A limited series expansion of this expression is obtained in this paper for $|z|>1$ as a product of convergent series.