Series

• Peter Leslie Walker

Abstract

A series is an expression of the form a1 + a2 + … + a n +…, where we begin with a given sequence (an) and form a new one by adding the terms together in order — then we say the series is convergent if the new sequence of sums is convergent in the sense of Chapter 1. For most of us this is the way in which we first meet the definitions and processes of analysis; for instance in what sense does the infinite sum $$1 - \frac{1} {2} + \frac{1} {4} + \frac{1} {8} + \cdot \cdot \cdot + \left( {\tfrac{{ - 1}} {2}} \right)^n + \cdot \cdot \cdot$$ , which we saw in the Preface, converge and why is its sum 2/3? And why does the sum $$1 - \frac{1} {2} + \frac{1} {3} - \frac{1} {4} + \cdot \cdot \cdot \frac{{\left( { - 1} \right)^{n - 1} }} {n} + \cdot \cdot \cdot$$ converge while the corresponding sum $$1 + \frac{1} {2} + \frac{1} {3} + \frac{1} {4} + \cdot \cdot \cdot \frac{1} {n} + \cdot \cdot \cdot$$ with positive terms does not? Our aim in this chapter is to put these ideas into a general framework and to derive the most commonly useful tests for convergence. For greater detail see for instance [1]

Keywords

Power Series Complex Number Cauchy Sequence Trigonometric Function Convergent Series
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