Some Simplifications in the Presentations of Complex Power Series and Unordered Sums

TL;DR

This paper offers simplified, accessible proofs of fundamental properties of complex power series by minimizing technical complexities and avoiding advanced mathematical tools, making the concepts easier to understand.

Contribution

It introduces simplified methods for presenting complex power series properties, reducing reliance on formal series, complex integration, and advanced analysis techniques.

Findings

Simplified proofs of addition, subtraction, multiplication, and division of power series

Clearer explanations of the principles of identity and isolated zeros

Easier derivation of Taylor's series and binomial series

Abstract

This text provides very easy and short proofs of some basic properties of complex power series (addition, subtraction, multiplication, division, rearrangement, composition, differentiation, uniqueness, Taylor's series, Principle of Identity, Principle of Isolated Zeros, and Binomial Series). This is done by simplifying the usual presentation of unordered sums of a (countable) family of complex numbers. All the proofs avoid formal power series, double series, iterated series, partial series, asymptotic arguments, complex integration theory, and uniform continuity. The use of function continuity as well as epsilons and deltas is kept to a minimum.