Series

A series is, informally speaking, the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely. Given an infinite sequence of numbers $\{ a_n \}$, a series is informally the result of adding all those terms together: $a_1 + a_2 + a_3 + \cdots$ . These can be written more compactly using the summation symbol $\Sigma$. An example is the famous series from Zeno's dichotomy and its mathematical representation:

$\displaystyle{\sum_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots}$

Zeno's Paradox

Say you are working from a location $x=0$ toward $x=100$. Before you can get there, you must get halfway there. Before you can get halfway there, you must get a quarter of the way there. Before traveling a quarter, you must travel one-eighth; before an eighth, one-sixteenth; and so on.

The terms of the series are often produced according to a certain rule, such as by a formula or by an algorithm. As there are an infinite number of terms, this notion is often called an infinite series. Unlike finite summations, infinite series need tools from mathematical analysis, specifically the notion of limits, 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, computer science, and finance.

Definition

For any sequence of rational numbers, real numbers, complex numbers, functions thereof, etc., the associated series is defined as the ordered formal sum:

$\displaystyle{\sum_{n=0}^{\infty}a_n = a_0 + a_1 + a_2 + \cdots}$

The sequence of partial sums $\{S_k\}$ associated to a series $\sum_{n=0}^\infty a_n$ is defined for each k as the sum of the sequence $\{a_n\}$ from $a_0$ to $a_k$:

$\displaystyle{S_k = \sum_{n=0}^{k}a_n = a_0 + a_1 + \cdots + a_k}$

By definition, the series $\sum_{n=0}^{\infty} a_n$ converges to a limit $L$ if and only if the associated sequence of partial sums $\{S_k\}$ converges to $L$. This definition is usually written as follows:

$\displaystyle{L = \sum_{n=0}^{\infty}a_n \Leftrightarrow L = \lim_{k \rightarrow \infty} S_k}$