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Series (mathematics)
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In mathematics, given an infinite sequence of numbers , a series is informally the result of adding all those terms together: . 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.
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In mathematics, given an infinite sequence of numbers , a series is informally the result of adding all those terms together: . 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 , define
Call the partial sum to N of the sequence , or partial sum of the series. A series is the sequence of partial sums, .
Potential confusion
When talking about series, one can refer either to the sequence of the partial sums, or to the sum of the series,
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i.e., the limit of the sequence of partial sums (see the formal definition in the next section) – it is clear which one is meant from context. To make a distinction between these two completely different objects (sequence vs. summed value), one sometimes omit the limits (atop and below the sum's symbol), as in
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in order to refer to the formal series, that may or may not have a definite sum.
Convergent series
A series ?an is said to 'converge' or to 'be convergent' when the sequence of partial sums has a finite limit. If the limit of SN is infinite or does not exist, the series is said to diverge. When the limit of partial sums exists, it is called the sum of the series
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The easiest way that an infinite series can converge is if all the an are zero for n sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.
Working out the properties of the series that converge even if all the terms are non-zero is the essence of the study of series. Consider the example
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, ½, ¼, 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 ½, we still have a piece of length ½ unmarked, so we can certainly mark the next ¼. 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.
Mathematicians extend this idiom to other, equivalent notions of series. For instance, when we talk about a recurring decimal, as in
we are talking, in fact, just about the series for which it stands (0.1 + 0.01 + 0.001 + …). But since these series always converge to real numbers (because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, it should offend no sensibilities if we make no distinction between 0.111… and 1/9. Less clear is the argument that , but it is not untenable when we consider that we can formalize the proof knowing only that limit laws preserve the arithmetic operations. See 0.999... for more.
Examples
- A geometric series is one where each successive term is produced by multiplying the previous term by a constant number. Example:
- In general, the geometric series
- converges if and only if |z| < 1.
- The harmonic series is divergent.
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- converges if r > 1 and diverges for r = 1, which can be shown with the integral criterion described below in convergence tests. As a function of r, the sum of this series is Riemann's zeta function.
- converges if the sequence bn converges to a limit L as n goes to infinity. The value of the series is then b1 − L.
Properties of series
Series are classed not only by whether they converge or diverge: they can also be split up based on the properties of the terms an (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term an (whether it is a real number, arithmetic progression, trigonometric function); etc.
Non-negative terms
When an is a non-negative real number for every n, the sequence SN of partial sums is non-decreasing. It follows that a series ?an with non-negative terms converges if and only if the sequence SN of partial sums is bounded.
For example, the series
is convergent, because the inequality
and a telescopic sum argument imply that the partial sums are bounded by 2.
Absolute convergence
A series
is said to converge absolutely if the series of absolute values
converges. It can be proved that this is sufficient to make not only the original series converge to a limit, but also for any reordering of it to converge to the same limit.
Conditional convergence
A series of real or complex numbers is said to be conditionally convergent (or semi-convergent) if it is convergent but not absolutely convergent. A famous example is the alternating series
which is convergent (and its sum is equal to ln 2), but the series formed by taking the absolute value of each term is the divergent harmonic series. The Riemann series theorem says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the an are real and S is any real number, that one can find a reordering so that the reordered series converges with sum equal to S.
Abel's test is an important tool for handling semi-convergent series. If a series has the form
where the partial sums BN = b0 + ··· + bn are bounded, ?n has bounded variation, and lim λnBn exists:
then the series ?an is convergent. This applies to the pointwise convergence of many trigonometric series, as in
with 0 < x < 2π. Abel's method consists in writing bn+1 = Bn+1 − Bn, and in performing a transformation similar to integration by parts (called summation by parts), that relates the given series ?an to the absolutely convergent series
Convergence tests
- n-th term test: If limn?8 an ? 0 then the series diverges.
- Comparison test 1: If ?bn is an absolutely convergent series such that |an | = C |bn | for some number C and for sufficiently large n , then ?an converges absolutely as well. If ?|bn | diverges, and |an | = |bn | for all sufficiently large n , then ?an also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the an alternate in sign).
- Comparison test 2: If ?bn is an absolutely convergent series such that |an+1 /an | = |bn+1 /bn | for sufficiently large n , then ?an converges absolutely as well. If ?|bn | diverges, and |an+1 /an | = |bn+1 /bn | for all sufficiently large n , then ?an also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the an alternate in sign).
- Ratio test: If |an+1/an| approaches a number less than one as n approaches infinity, then ?an converges absolutely. When the limit of the ratio is 1, convergence can sometimes be determined as well.
- Root test: If there exists a constant C < 1 such that |an|1/n = C for all sufficiently large n, then ?an converges absolutely.
- Integral test: if ƒ(x) is a positive monotone decreasing function defined on the interval [1, 8) with ƒ(n) = an for all n, then ?an converges if and only if the integral ?18 ƒ(x) dx is finite.
- Cauchy's condensation test: If an is non-negative and non-increasing, then the two series ?an and ?2ka(2k) are of the same nature: both convergent, or both divergent.
- Alternating series test: A series of the form ?(−1)n an (with an = 0) is called alternating. Such a series converges if the sequence an 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.
Series of functions
A series of real- or complex-valued functions
converges pointwise on a set E, if the series converges for each x in E as an ordinary series of real or complex numbers. Equivalently, the partial sums
converge to ƒ(x) as N → 8 for each x ∈ E.
A stronger notion of convergence of a series of functions is called uniform convergence. The series converges uniformly if it converges pointwise to the function ƒ(x), and the error in approximating the limit by the Nth partial sum,
can be made small independently of x by choosing a sufficiently large N.
Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit. For example, if a series of continuous functions converges uniformly, then the limit function is also continuous. Similarly, if the ƒn are integrable on a closed and bounded interval I and converge uniformly, then the series is also integrable on I and can be integrated term-by-term. Tests for uniform convergence include the Weierstrass' M-test, Abel's uniform convergence test, Dini's test, and the Cauchy criterion.
More sophisticated types of convergence of a series of functions can also be defined. In measure theory, for instance, a series of functions converges almost everywhere if it converges pointwise except on a certain set of measure zero. Other modes of convergence depend on a different metric space structure on the space of functions under consideration. For instance, a series of functions converges in mean on a set E to a limit function ƒ provided
as N → 8.
Power series
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Many functions can be represented as Taylor series; these are infinite series involving powers of the independent variable and are also called power series. For example, the series
converges to for all x.
In general, a power series is any series of the form
Such a series converges on a certain open disc of convergence centered at the point c, and may also converge at some of the points of the boundary. The radius of this disc is known as the radius of convergence, and can in principle be determined from the asymptotics of the coefficients an. The convergence is uniform on closed and bounded (that is, compact) subsets of the interior of the disc of convergence: to wit, it is uniformly convergent on compact sets.
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.
Laurent series
Laurent series generalize power series by admitting terms into the series with negative as well as positive exponents. A Laurent series is thus any series of the form
If such a series converges, then in general it does so in an annulus rather than a disc, and possibly some boundary points. The series converges uniformly on compact subsets of the interior of the annulus of convergence.
Dirichlet series
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A Dirichlet series is one of the form
where s is a complex number. For example, if all an are equal to 1, then the Dirichlet series is the Riemann zeta function
Like the zeta function, Dirichlet series in general play an important role in analytic number theory. Generally a Dirichlet series converges if the real part of s is greater than a number called the abscissa of convergence. In many cases, a Dirichlet series can be extended to an analytic function outside the domain of convergence by analytic continuation. For example, the Dirichlet series for the zeta function converges absolutely when Re s > 1, but the zeta function can be extended to a holomorphic function defined on with a simple pole at 1.
Trigonometric series
A series of functions in which the terms are trigonometric functions is called a trigonometric series:
The most important example of a trigonometric series is the Fourier series of a function.
History of the theory of infinite series
Development of infinite series
Greek mathematician Archimedes produced the first known summation of an infinite series with a
method that is still used in the area of calculus today. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of π.
The idea of an infinite series expansion of a function was first conceived in India by Madhava in the 14th century, who also developed precursors to the modern concepts of the power series, the Taylor series, the Maclaurin series, rational approximations of infinite series, and infinite continued fractions. He discovered a number of infinite series, including the Taylor series of the trigonometric functions of sine, cosine, tangent and arctangent, the Taylor series approximations of the sine and cosine functions, and the power series of the radius, diameter, circumference, angle ?, p and p/4. His students and followers in the Kerala School further expanded his works with various other series expansions and approximations, until the 16th century.
In the 17th century, James Gregory also worked on infinite series and published several Maclaurin series. In 1715, a general method for constructing the Taylor series for all functions for which they exist was provided by Brook Taylor. Leonhard Euler in the 18th century, developed the theory of hypergeometric series and q-series.
Convergence criteria
The study of the convergence criteria of a series began with Madhava in the 14th century, who developed tests of convergence of infinite series, which his followers further developed at the Kerala School.
In Europe, however, the investigation of the validity of infinite series is considered to begin with Gauss in the 19th century. Euler had already considered the hypergeometric series
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). Leonhard Euler and Gauss had given various criteria, and Colin 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 binomial series
corrected certain of Cauchy's conclusions, and gave a completely
scientific summation of the series for complex values of and . 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), Chebyshev (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 Seidel and Stokes (1847-48). Cauchy took up the
problem again (1853), acknowledging Abel's criticism, and reaching
the same conclusions which Stokes had already found. Thomae 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
A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not absolutely convergent.
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
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 x in terms of the sines or cosines of
multiples of x, 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.
Generalizations
Asymptotic series
Asymptotic series, otherwise asymptotic expansions, are infinite series whose partial sums become good approximations in the limit of some point of the domain. In general they do not converge. But they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical asymptotic series reaches its best approximation; if more terms are included, most such series will produce worse answers.
Divergent series
Under many circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense. A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence. Summability methods include Cesàro summation, (C,k) summation, Abel summation, and Borel summation, in increasing order of generality (and hence applicable to increasingly divergent series).
A variety of general results concerning possible summability methods are known. The Silverman–Toeplitz theorem characterizes matrix summability methods, which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients. The most general method for summing a divergent series is non-constructive, and concerns Banach limits.
Series in Banach spaces
The notion of series can be easily extended to the case of a Banach space. If xn is a sequence of elements of a Banach space X, then the series Σxn converges to x ∈ X if the sequence of partial sums of the series tends to x; to wit,
as N → 8.
More generally, convergence of series can be defined in any abelian Hausdorff topological group. Specifically, in this case, Σxn converges to x if the sequence of partial sums converges to x.
Summations over arbitrary index sets
Definitions may be given for sums over an arbitrary index set I. There are two main differences with the usual notion of series: first, there is no specific order given on the set I; second, this set I may be uncountable.
Families of non-negative numbers
When summing a family , i ∈ I, of non-negative numbers, one may define
When the sum is finite, the set of i ∈ I such that ai > 0 is countable. Indeed for every n ≥ 1, the set is finite, because
If I is countably infinite and enumerated as I = then the above defined sum satisfies
provided the value ∞ is allowed for the sum of the series.
Any sum over non-negative reals can be understood as the integral of a non-negative function with respect to the counting measure, which accounts for the many similarities between the two constructions.
Abelian topological groups
Let a : I ? X, where I is any set and X is an abelian Hausdorff topological group. Let F be the collection of all finite subsets of I. Note that F is a directed set ordered under inclusion with union as join. Define the sum S of the family a as the limit
if it exists and say that the family a is unconditionally summable. Saying that the sum S is the limit of finite partial sums means that for every neighborhood V of 0 in X, there is a finite subset A0 of I such that
Because F is not totally ordered, this is not a limit of a sequence of partial sums, but rather of a net.
For every W, neighborhood of 0 in X, there is a smaller neighborhood V such that V − V ⊂ W. It follows that the finite partial sums of an unconditionally summable family ai, i ∈ I, form a Cauchy net, that is: for every W, neighborhood of 0 in X, there is a finite subset A0 of I such that
When X is complete, a family a is unconditionally summable in X if and only if the finite sums satisfy the latter Cauchy net condition. When X is complete and ai, i ∈ I, is unconditionally summable in X, then for every subset J ⊂ I, the corresponding subfamily aj, j ∈ J, is also unconditionally summable in X.
When the sum of a family of non-negative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological group X = R.
If a family a in X is unconditionally summable, then for every W, neighborhood of 0 in X, there is a finite subset A0 of I such that ai ∈ W for every i not in A0. If X is first-countable, it follows that the set of i ∈ I such that ai ≠ 0 is countable. This need not be true in a general abelian topological group (see examples below).
Unconditionally convergent series
Suppose that I = N. If a family an, n ∈ N, is unconditionally summable in an abelian Hausdorff topological group X, then the series in the usual sense converges and has the same sum,
By nature, the definition of unconditional summability is insensitive to the order of the summation. When ?an is unconditionally summable, then the series remains convergent after any permutation s of the set N of indices, with the same sum,
It can be proved that the converse holds: is a series ?an converges after any permutation, then it is unconditionally convergent. When X is complete, then unconditional convergence is also equivalent to the fact that all subseries are convergent; if X is a Banach space, this is equivalent to say that for every sequence of signs en = 1 or −1, the series
converges in X. If X is a Banach space, then one may define the notion of absolute convergence. A series ?an of vectors in X converges absolutely if
If a series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite dimensional Banach spaces (theorem of ).
Well-ordered sums
Conditionally convergent series can be considered if I is a well-ordered set, for example an ordinal α0. One may define by transfinite induction
and for a limit ordinal α,
if this limit exists. If all limits exist up to α0, then the series converges.
Examples
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Given a function f : X?Y, with Y an abelian topological group, define for every a ∈ X
the function whose support is a singleton . Then
in the topology of pointwise convergence (that is, the sum is taken in the infinite product group YX ).
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In the definition of partitions of unity, one constructs sums of functions over arbitrary index set I,
While, formally, this requires a notion of sums of uncountable series, by construction there are, for every given x, only finitely many nonzero terms in the sum, so issues regarding convergence of such sums do not arise. Actually, one usually assumes more: the family of functions is locally finite, i.e., for every x there is a neighborhood of x in which all but a finite number of functions vanish. Any regularity property of the fi, such as continuity, differentiability, that is preserved under finite sums will be preserved for the sum of any subcollection of this family of functions.
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On the first uncountable ordinal ω1 viewed as a topological space in the order topology, the constant function f: [0,?1) ? [0,?1] given by f(a) = 1 satisfies
(in other words, ?1 copies of 1 is ?1) only if one takes a limit over all countable partial sums, rather than finite partial sums. This space is not separable.
See also
External links
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