Series (mathematics)

In mathematics, a series is the sum of a sequence of terms. That is, a series is a list of numbers with addition operations between them, e.g,

1 + 2 + 3 + 4 + 5 + ...

which may or may not be meaningful.

In most cases of interest the terms of the sequence are produced according to a certain rule, e.g., by a formula, by an algorithm, by a sequence of measurements, or even by a random number generator.

Series may be finite, or infinite; in the first case they may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way.

Examples of simple series include the arithmetic series which is a sum of an arithmetic progression, written as:

and finite geometric series, a sum of a geometric progression, which can be written as:

Contents

1 Infinite series

1.1 Formal definition

2 History of the theory of infinite series

2.1 Convergence criteria
2.2 Uniform convergence
2.3 Semi-convergence
2.4 Fourier series

3 Some types of infinite series

4 Convergence tests

5 Examples

6 Absolute convergence

7 Power series

8 Generalizations

9 See also

Infinite series

The sum of an infinite series is a limit of partial sums of infinitely many terms. Such a limit can have a finite value; if it has, the series is said to converge; if it does not, it is said to diverge. The fact that infinite series can converge resolves several of Zeno's paradoxes.

The simplest convergent infinite series is perhaps

<math>1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots<math>

It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2 — in other words, the series has an upper bound.

This series is a geometric series and mathematicians usually write it as:

<math>\sum_{n=0}^\infty 2^{-n}=2.<math>

An infinite series is formally written as

<math>\sum_{n=0}^\infty a_n<math>

where the elements a_n are real (or complex) numbers. We say that this series converges towards S, or that its value is S, if the limit

<math>\lim_{N\rightarrow\infty}\sum_{n=0}^N a_n<math>

exists and is equal to S. If there is no such number, then the series is said to diverge.

The sequence of partial sums is defined as the sequence

indexed by N. Then, the definition of series convergence simply says that the sequence of partial sums has limit S, as N → ∞.

Formal definition

Indeed, mathematicians usually define a series as the above sequence of partial sums. The notation <math>\sum_{n=0}^\infty a_n<math> represents then a priori this sequence, which is always well defined, but which may or may not converge. Only in the latter case, i.e. if this sequence has a limit, the notation is also used to denote the limit of this sequence. To make a distinction between these two completely different objects (sequence vs. numerical value), one may omit the limits (atop and below the sum's symbol) in the former case.

Also, different notions of convergence of such a sequence do exist (absolute convergence, summability...). In case the elements of the sequence (and thus of the series) are not simple numbers, but e.g. functions, still more types of convergence can be considered (pointwise convergence, uniform convergence (see below)).

History of the theory of infinite series

Convergence criteria

The investigation of the validity of infinite series is considered to begin with Gauss. Euler had already considered the hypergeometric series

<math>1 + \frac{\alpha\beta}{1\cdot\gamma}x + \frac{\alpha(\alpha+1)\beta(\beta+1)}{1 \cdot 2 \cdot \gamma(\gamma+1)}x^2 + \cdots.<math>

on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.

Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms convergence and divergence had been introduced long before by Gregory (1668). Euler and Gauss had given various criteria, and Maclaurin had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of power series by his expansion of a complex function in such a form.

Abel (1826) in his memoir on the series

<math>1 + \frac{m}{1}x + \frac{m(m-1)}{2!}x^2 + \cdots<math>

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of <math>m<math> and <math>x<math>. He showed the necessity of considering the subject of continuity in questions of convergence.

Cauchy's methods led to special rather than general criteria, and the same may be said of Raabe (1832), who made the first elaborate investigation of the subject, of De Morgan (from 1842), whose logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have shown to fail within a certain region; of Bertrand (1842), Bonnet (1843), Malmsten (1846, 1847, the latter without integration); Stokes (1847), Paucker (1852), Tchebichef (1852), and Arndt (1853).

General criteria began with Kummer (1835), and have been studied by Eisenstein (1847), Weierstrass in his various contributions to the theory of functions, Dini (1867), DuBois-Reymond (1873), and many others. Pringsheim's (from 1889) memoirs present the most complete general theory.

Uniform convergence

The theory of uniform convergence was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it successfully were Stokes and Seidel (1847-48). Cauchy took up the problem again (1853), acknowledging Abel's criticism, and reaching the same conclusions which Stokes had already found. Thomé used the doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform convergence, in spite of the demands of the theory of functions.

Semi-convergence

Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by Malmsten (1847). Schlömilch (Zeitschrift, Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function <math>F(x) = 1^n + 2^n + \cdots + (x - 1)^n<math>. Genocchi (1852) has further contributed to the theory.

Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until Cayley (1873) brought it into prominence.

Fourier series

Fourier series were being investigated as the result of physical considerations at the same time that Gauss, Abel, and Cauchy were working out the theory of infinite series. Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète. Euler and Lagrange simplified the subject, as did Poinsot, Schröter, Glaisher, and Kummer.

Fourier (1807) set for himself a different problem, to expand a given function of <math>x<math> in terms of the sines or cosines of multiples of <math>x<math>, a problem which he embodied in his Théorie analytique de la Chaleur (1822). Euler had already given the formulas for determining the coefficients in the series; Fourier was the first to assert and attempt to prove the general theorem. Poisson (1820-23) also attacked the problem from a different standpoint. Fourier did not, however, settle the question of convergence of his series, a matter left for Cauchy (1826) to attempt and for Dirichlet (1829) to handle in a thoroughly scientific manner (see convergence of Fourier series). Dirichlet's treatment (Crelle, 1829), of trigonometric series was the subject of criticism and improvement by Riemann (1854), Heine, Lipschitz, Schläfli, and DuBois-Reymond. Among other prominent contributors to the theory of trigonometric and Fourier series were Dini, Hermite, Halphen, Krause, Byerly and Appell.

Some types of infinite series

A geometric series is one where each successive term is produced by multiplying the previous term by a constant number. Example:

<math>1 + {1 \over 2} + {1 \over 4} + {1 \over 8} + {1 \over 16} + \cdots=\sum_{n=0}^\infty{1 \over 2^n}.<math>

The harmonic series is the series

<math>1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + {1 \over 5} + \cdots =\sum_{n=1}^\infty 1/n.<math>

An alternating series is a series where terms alternate signs. Example:

<math>1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots =\sum_{n=1}^\infty (-1)^{n+1} {1 \over n}.<math>

Convergence tests

Comparison test 1: If ∑b_n is an absolutely convergent series such that |a_n | ≤ C |b_n | for some number C and for sufficiently large n , then ∑a_n converges absolutely as well. If ∑|b_n | diverges, and |a_n | ≥ |b_n | for all sufficiently large n , then ∑a_n also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n alternate in sign).
Comparison test 2: If ∑b_n is an absolutely convergent series such that |a_n+1 /a_n | ≤ C |b_n+1 /b_n | for some number C and for sufficiently large n , then ∑a_n converges absolutely as well. If ∑|b_n | diverges, and |a_n+1 /a_n | ≥ |b_n+1 /b_n | for all sufficiently large n , then ∑a_n also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n alternate in sign).
Ratio test: If |a_n+1/a_n| < 1 for all sufficiently large n, then ∑ a_n converges absolutely. When the ratio is 1, convergence can sometimes be determined as well.
Root test: If there exists a constant C < 1 such that |a_n|^1/n ≤ C for all sufficiently large n, then ∑ a_n converges absolutely.
Integral test: if f(x) is a positive monotone decreasing function defined on the interval [1, ∞) with f(n) = a_n for all n, then ∑ a_n converges if and only if the integral ∫₁^∞ f(x) dx is finite.
Alternating series test: A series of the form ∑ (−1)ⁿ a_n (with a_n ≥ 0) is called alternating. Such a series converges if the sequence a_n is monotone decreasing and converges to 0. The converse is in general not true.
For some specific types of series there are more specialized convergence tests, for instance for Fourier series there is the Dini test.

Examples

The series

<math>\sum_{n=1}^\infty\frac{1}{n^r}<math>

converges if r > 1 and diverges for r ≤ 1, which can be shown with the integral criterion 5) from above. As a function of r, the sum of this series is Riemann's zeta function.

The geometric series

<math>\sum_{n=0}^\infty z^n<math>

converges if and only if |z| < 1.

The telescoping series

<math>\sum_{n=1}^\infty (b_n-b_{n+1})<math>

converges if the sequence b_n converges to a limit L as n goes to infinity. The value of the series is then b₁ − L.

The power series of <math>e^x<math>

<math>\sum_{n=0}^\infty\frac{x^n}{n!}<math>

converges to <math>e^x<math>.

Absolute convergence

Main article: absolute convergence.

The sum

<math>\sum_{n=0}^\infty a_n<math>

is said to converge absolutely if the series of absolute values

<math>\sum_{n=0}^\infty \left|a_n\right|<math>

converges. In this case, the original series, and all reorderings of it, converge, and converge towards the same sum.

The Riemann series theorem says that if a series converges, but not absolutely, then one can always find a reordering of the terms so that the reordered series diverges. Moreover, if the a_n are real and S is any real number, one can find a reordering so that the reordered series converges with limit S.

Power series

Several important functions can be represented as Taylor series; these are infinite series involving powers of the independent variable and are also called power series. See also radius of convergence.

Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required. However, the formal operation with non-convergent series has been retained in rings of formal power series which are studied in abstract algebra. Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions.

Generalizations

The notion of series can be defined in every abelian topological group; the most commonly encountered case is that of series in a Banach space.

There is no serious definition for an infinite sum over an uncountable set. For example if X is a set and f a function on X taking non-negative real values, such that

<math>\sum_{y\in Y} f(y)

for any countable subset Y of X, with A an absolute constant, it follows that f(x) = 0 for all x outside some countable subset of X. In other words, infinite sums of uncountably many non-negative reals make sense only in the case that this is a conventional convergent infinite series, extended by the value 0 to an uncountable set.

Asymptotic series, otherwise asymptotic expansions, are not typically convergent infinite series, but sequences of finite approximations each of which is a good asymptotic representation.