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

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.