In mathematics, given an infinite sequence of numbers { a_n }, a series is informally the result of adding all those terms together: a₁ + a₂ + a₃ + ·. These can be written more compactly using the summation symbol ∑. An example is the famous series from Zeno's dichotomy

The terms of the series are often produced according to a certain rule, such as by a formula, by an algorithm, by a sequence of measurements, or even by a random number generator. As there are an infinite number of terms, this notion is often called an infinite series. Unlike finite summations, series need tools from mathematical analysis to be fully understood and manipulated. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics and computer science.

Basic properties

Series can be composed of terms from any one of many different sets including real numbers, complex numbers, and functions. The definition given here will be for real numbers, but can be generalized.

Given an infinite sequence of real numbers { a_n }, define

Call S_N the partial sum to N of the sequence { a_n }, or partial sum of the series. A series is the sequence of partial sums, { S_N }.