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

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.

💡 Research Summary

The paper addresses a classical limitation in the binomial expansion of the function ((1+z)^A) when the exponent (A) is a non‑zero real number that is not a positive integer. It is well known that the standard Taylor (or binomial) series ((1+z)^A=\sum_{n=0}^{\infty}\binom{A}{n}z^{n}) converges only for (|z|<1). Consequently, for (|z|>1) the series diverges, leaving a gap in analytic representations that are needed in many areas of physics and engineering where the argument may lie outside the unit disc.

The authors begin by recalling the derivation of the classical binomial series and explicitly demonstrating why the radius of convergence is exactly one when (A) is non‑integral. They then introduce a simple but powerful algebraic manipulation: rewrite the function as
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