Series (mathematical sums, convergence, and applications)

A series is the sum of terms taken in sequence; in mathematics it often means an infinite sum. Topics include notation, convergence and divergence, common examples, tests, history, and applications.

Overview

A series is, in its broadest sense, a collection of similar items arranged and considered together because they follow a common rule or theme. In mathematics the term normally refers to the sum of the terms of a sequence. A finite series is simply the sum of a fixed number of terms; an infinite series continues without end and is studied by examining its sequence of partial sums. Notation commonly uses the summation symbol ∑ to indicate adding successive terms, and the concept of a limit is central to deciding whether an infinite series has a well-defined total. More on mathematical series $\textstyle \sum _{n=i}^{k}a_{n}$

Definition and basic properties

Formally, given a sequence (a_n) of numbers (or other mathematical objects in some contexts), the series is the expression a_1 + a_2 + a_3 + ⋯. For an infinite series one considers the n-th partial sum S_n = a_1 + a_2 + ... + a_n. If the sequence (S_n) approaches a finite limit S as n→∞, the series is said to converge and S is its sum. If (S_n) fails to approach a finite limit—for example it grows without bound or oscillates—the series diverges. The idea of a limit and convergence is standard in analysis and is discussed further in introductory texts on limits and sequences. Definition of limit $a_{i}+\cdots +a_{k}$

Common examples and intuition

Two classical examples illustrate contrasting behavior. A geometric series with first term a and common ratio r, written a + ar + ar^2 + ar^3 + ⋯, converges to a/(1−r) when |r| < 1 and diverges otherwise; this provides a simple model for many decaying processes and for summation techniques. By contrast, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ⋯ diverges even though its terms tend to zero; this shows that having terms that vanish is necessary but not sufficient for convergence. Partial sums of convergent series often approach their limit rapidly in geometric cases but much more slowly for series like the harmonic one. $\textstyle \sum _{n=0}^{\infty }{\frac {1}{2^{n}}}$ $1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}}+{\frac {1}{128}}+\ldots$

Tests for convergence

Over time mathematicians developed practical tests to determine whether a given infinite series converges. Useful and commonly taught tests include:

Comparison test: compare with a known convergent or divergent series.
Ratio test: examine the limit of |a_{n+1}/a_n| for large n.
Root test: examine the n-th root of |a_n| as n→∞.
Integral test: relate the series to an improper integral when terms come from a decreasing positive function.
Alternating series test (Leibniz criterion): applies when signs alternate and magnitudes decrease to zero.

Each test has conditions under which it is decisive; often several tests are used together to arrive at a conclusion. $1+{\frac {1}{2}}=1{\frac {1}{2}}$ $1+{\frac {1}{2}}+{\frac {1}{4}}=1{\frac {3}{4}}$ $1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}=1{\frac {7}{8}}$

History and development

The study of infinite series dates back to ancient and early modern mathematics, but it was in the 17th and 18th centuries that rigorous ideas about convergence were formulated. Work by mathematicians such as Newton, Leibniz, Euler and Cauchy clarified when infinite sums could be manipulated and when they represented meaningful quantities. The modern theory of series is embedded in real and complex analysis and underlies important constructions like power series and Fourier series. These developments made it possible to represent functions by infinite sums and to analyze differential equations and physical phenomena. $1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}=1{\frac {15}{16}}$ $1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}=1{\frac {31}{32}}$

Applications, distinctions, and notable facts

Series are central across mathematics and its applications. Power series allow functions to be expressed as sums of powers of a variable and provide local approximations; Fourier series decompose periodic signals into sines and cosines and are fundamental in engineering, physics and signal processing. In numerical work, series can be used to obtain high-precision values for constants and functions, but care is required: rearranging terms in a conditionally convergent series can change its sum, while absolutely convergent series are robust under rearrangement. A series should not be confused with a sequence: a sequence lists terms, whereas a series refers to their sum. For further reading about examples that diverge or behave unexpectedly, see discussions of alternating and conditionally convergent series as well as the classical harmonic series. Divergence and infinity • Harmonic series $1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}}=1{\frac {63}{64}}$ $1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}}+{\frac {1}{128}}=1{\frac {127}{128}}$