Infinite Series

Abstract

The axioms for the real numbers define addition as a binary operation and establish the rules for adding two real numbers together. One can use mathematical induction to extend axioms and theorems about addition to get theorems about the addition of any finite number of terms. But there is nothing in the axioms that suggests how to add an infinite number of terms together or what such a sum would mean. You need to make a separate definition in order to make sense out of adding infinitely many terms together. An infinite series $a_{1} + a_{2} + a_{3} + \cdots =\sum \limits _{ n=1}^{\infty }a_{n}$ has a sequence of terms $a_{1},a_{2},a_{3},\ldots$ which are written with plus signs or minus signs between the terms of the sequence. In this chapter, most series will begin with a first term a ₁, although there is no problem with beginning the series at other subscript values such as the commonly seen $\sum \limits _{n=0}^{\infty }a_{n}$. Also in this chapter the terms of the series will be real numbers, although it is possible to extend the definition to series of other kinds of terms such as complex numbers or matrices. This explains what an infinite series looks like, but it does not prescribe any meaning to the symbols.