A sequence of elements (called the terms of the given series) of some linear topological space and a certain infinite set of their partial sums (called the partial sums of the series) for which the notion of a limit is defined. Here are the simplest examples of series.
Simple series of numbers.
A pair of sequences of complex numbers and such that
is called a (simple) series of numbers and is denoted as follows:
The elements of the sequence are called the terms of the series and the elements of are called its partial sums; moreover, is called the -th term of the series (2), and its partial sum of order . The series (2) is defined uniquely by each of the two sequences and : the terms of the sequence are obtained from the terms of the sequence by formula (1), and the sequence can be recovered from by the formulas
From this point of view the study of series is equivalent to the study of sequences: For any statement about series one can formulate an equivalent statement about sequences.
A series (2) is called convergent if the sequence of its partial sums has a finite limit
which is called the sum of the series (2) and is written as
Thus, the notation (2) is used both for the series itself and for its sum. If the sequence of partial sums of the series (2) does not have a finite limit, then the series is called divergent (cf. also Divergent series).
An example of a convergent series is the sum of the terms of an infinite geometric progression
provided that . In this case its sum is equal to , i.e. . If , (3) is an example of a divergent series.
If the series (2) is convergent, then the sequence of terms tends to zero:
The converse of this statement is not true: The sequence of terms of the harmonic series
tends to zero though this series is divergent.
The series is called the remainder of order of the series (2). If a series is convergent, then each remainder of it is convergent. If some remainder of a series is convergent, then the series itself is convergent. If the remainder of order of the series (2) is convergent and its sum is equal to , that is, , then
If the series (2) and the series
are convergent, then the series
is also convergent; this series is called the sum of the series (2) and (4); moreover, its sum is equal to the sum of these series.
If the series (2) is convergent and is a complex number, then the series , called the product of the series (2) and the number , is also convergent, and .
A condition for the convergence of a series which does not use the notion of its sum is the Cauchy criterion for the convergence of a series.
If all terms of the series (2) are real numbers, , then the series (2) is called real. In the theory of series an important role is played by real series with non-negative terms:
A necessary and sufficient condition for the convergence of the series (5) is that the sequence of its partial sums is bounded above. If this series is divergent, then its partial sums tend to infinity:
therefore, in this case one writes
For series with non-negative terms there exist quite a number of convergence criteria. The following criteria are the principal ones.
The comparison test. If for a series (5) and for a series
with non-negative terms there exists a constant such that , then the convergence of the series (6) implies the convergence of the series (5), and the divergence of (5) implies the divergence of (6).
When the comparison test is applied in studies of the convergence for a given series with non-negative terms, it is often reasonable to single out the principal part of its -th term with respect to as in the form ( is some constant), and to take the series
as a comparison series. This series is convergent for and divergent for .
The rule below follows from the comparison test in case one takes the series (7) as a comparison series: If
then for and the series (5) converges, and for and the series (5) diverges.
The comparison test also implies the d'Alembert criterion (convergence of series) and the Cauchy criterion for the convergence of a series of positive numbers. For such series there also exist the criteria of Bertrand, Gauss, Ermakov, Kummer, and Raabe (cf. Bertrand criterion; Gauss criterion; Ermakov convergence criterion; Kummer criterion; and Raabe criterion).
The integral test for convergence provides sufficient conditions for the convergence of a series (5) with non-negative terms forming a decreasing sequence: , . Let a series (5) be such that there exists a function , defined and decreasing for , the values of which on the integers coincide with the terms of the given series: , . Then, if are the partial sums and are the remainders of (5), the following estimates are valid:
where is some constant and .
Thus, the series (5) converges if and only if the integral is convergent.
If the series (5) is divergent, then its partial sums increase in the same way as the integrals , i.e. the are asymptotically equal to the indicated integrals:
For a series (5) whose terms form a decreasing sequence the following Cauchy condensation theorem is valid: If the terms of (5) decrease, then it converges or diverges simultaneously with the series
A necessary condition for the convergence of a series (5) with a decreasing sequence of terms is the condition
The example of the divergent series
shows that condition (8) is not sufficient for the convergence of a series (5) with a decreasing sequence of terms.
An important class of series of numbers are the absolutely convergent series, i.e. series (2) for which the series are convergent. If a series is absolutely convergent, then it is convergent and its sum is independent of the order in which the summands are written. Series that are convergent but not absolutely convergent are called conditionally convergent. An example of a conditionally convergent series is the series
The sum of a conditionally convergent series depends on the order in which its terms are written (see Riemann theorem on the rearrangement of the terms of a series): Whatever and belonging to the set of real numbers completed by the infinities and , , one can rearrange the terms of any conditionally convergent series with real numbers as its terms so that for the partial sums of the resulting series the following equalities will hold:
Thus, for conditionally convergent series the commutative law of addition is not valid. Also, the associative law of addition does not hold for all series: If a series is divergent, then a series obtained from it by a sequential grouping of terms can be convergent; moreover, its sum depends on the way of grouping the terms of the original series. For example, the series
is divergent, but the series and obtained from it by pairwise grouping of its terms are convergent and have different sums. However, if a series is convergent, then, of course, any series obtained from it by a sequential grouping of its terms is convergent and its sum is the sum of the given series, since the sequence of partial sums of the new series is a subsequence of the sequence of partial sums of the original series.
Among the series with terms of different signs it is usual to single out the alternating series for which the Leibniz criterion for convergence is valid. Different criteria for the convergence of arbitrary series of numbers can be obtained by the Abel transformation of the sums of pairwise products, for example, the Abel criterion; the Dedekind criterion (convergence of series); the Dirichlet criterion (convergence of series); and the du Bois-Reymond criterion (convergence of series).
Multiplication of series. There are different rules for the multiplication of series. The best known is Cauchy's rule, according to which to multiply two series (2) and (4) one sums at first in finite "diagonals" the pairwise products , i.e. the products in which the sum of indices has the same value:
and the series for which the obtained sums are the terms is called the Cauchy product of the two given series. This rule of multiplication of series is suggested by the formula for multiplication of power series:
Let the series (2), (4) and (9) be convergent and let
If the series (2) and (4) are absolutely convergent, then the series (9) is also absolutely convergent and . If the series (2) is absolutely convergent and the series (4) is convergent, then (9) is convergent and (Mertens' theorem). If the series (2) and (4) are conditionally convergent, then (9) may be divergent; for example, the series
is conditionally convergent and the series
is divergent (its terms do not tend to 0). If all three series — (2), (4) and (9) — are convergent, then (Abel's product theorem).
An example of another rule of multiplication of series is the rule in which at first one carries out the summation of the pairwise products in which the product of the indices has a fixed value:
then the product of the series (2) and (4) is defined as the series . This rule of multiplication is suggested by the formula for multiplication of Dirichlet series:
There also exist series with terms numbered by all integers . They are denoted by
A series (10) is called convergent if the series
are both convergent and the sum of the sums of these two series is called the sum of (10).
Series of numbers of a more complicated structure are multiple series, which have terms provided with multi-indices, where the are positive integers, ; . In the theory of multiple series various types of partial sums are considered: triangular
and others. According to the chosen type of partial sums one can define the notion of the sum of a multiple series as their corresponding limit. In the case when , a multiple series is called a double series. For multiple series, unlike simple series, the given set of partial sums does not determine the terms of a series, i.e. in general, to define a multiple series it is necessary that both the multiple sequence of its terms and the set of its partial sums be given.
In mathematical analysis both convergent and divergent series are used. For the latter various methods of summation are worked out.
Many important irrational constants can be obtained as sums of series of numbers, for example:
the same is true for the values of definite integrals in which the primitives of the integrands cannot be written in elementary functions:
Series of functions.
A (simple) series of functions
is a pair of sequences of functions and consisting of numerical functions defined on some set and such that
As in the case of series of numbers, the elements of the sequence are called the terms of the series (11) and those of the sequence — its partial sums. The series (11) is called convergent on the set if for each fixed the following series of numbers is convergent:
Example. The series
is convergent on the entire complex plane and the series
only when .
The sum of a convergent series of functions continuous, for example, on some interval is not necessarily a continuous function; for example, the series
is convergent on the interval , its terms are continuous on this interval but the sum
is discontinuous at the point . A number of conditions under which the properties of continuity, differentiability and integrability of finite sums of, respectively, continuous, differentiable or integrable functions can be carried over to series of functions are formulated in terms of uniform convergence of a series (see Uniformly-convergent series).
Series of measurable functions. Let be a Lebesgue-measurable subset of the -dimensional Euclidean space , let be the Lebesgue measure and let the terms of the series
be measurable, almost-everywhere finite functions on taking values in the extended real line (i.e. together with real numbers they can assume the values and ). If the series (12) is convergent almost-everywhere on , then its sum is also a measurable function, and by Egorov's theorem (cf. Egorov theorem), if , then for any there exists a compact set such that and such that the series with as terms the restrictions of the functions to converges uniformly on and its sum is — the restriction of the sum (12) to .
Let , , be the space of functions with the norm
for and with the norm
for , respectively. A series (12) is called convergent in if the sequence of its partial sums converges in , and its limit is called the sum of (12) in this space:
If a series (12) converges in and is its sum, then there exists a subsequence of the sequence of its partial sums which converges to almost-everywhere on .
Term-by-term integration of series. The following theorems are extensions of the theorem on term-by-term integration of uniformly-convergent series.
Theorem 1. If there exists a summable function on the set such that for all and all the partial sums of the series (12) satisfy the inequality
if the series (12) converges almost-everywhere on and if its sum is , then
Theorem 2. If , , , , and if the sequence of partial sums of the series (12) converges weakly to the function (i.e. for any function , , the following condition is satisfied:
then formula (13) holds.
One can also carry out term-by-term integration of a series (12) all terms of which are non-negative on the set . For such a series the sequence of their partial sums at each point increases and thus has a finite or infinite limit, which is called the value of the sum of the series at this point.
Theorem 3. If the terms of (12) are non-negative, then formula (13) holds.
Under the assumptions of theorem 3 both sides of formula (13) can be . The following theorem provides sufficient conditions for their finiteness.
Theorem 4. If the terms of (12) are non-negative and if the integrals of their partial sums are uniformly bounded:
where is a constant, then the sum of the series (12) is a summable function.
Term-by-term differentiation of series. Let be the -dimensional Euclidean space of points , , let be an open set in , and . Let be the generalized derivative of the function with respect to , . If , ( is fixed), , and if the series and , are convergent in : and , then has a generalized derivative in with respect to and , i.e.
Among series of functions, especially important are power series; Fourier series; Dirichlet series, and, in general, series obtained by the expansion of functions in terms of the eigenfunctions of some operator. Many of the stated properties of series of functions can be extended to more general series with terms which are functions with values in linear normed spaces or, more generally, in linear topological spaces, and also to multiple series of functions, i.e. series whose terms are provided with multi-indices:
The theory of series of functions provides convenient and quite general methods for studying functions, since a rather wide class of functions can be represented in a certain sense as the sum of a series of elementary functions. For example, a single-valued analytic function is the sum of its Taylor series in a neighbourhood of each interior point of its domain of definition; any continuous function on some interval is the sum of a series converging uniformly on this interval with algebraic polynomials as terms; finally, for any measurable almost-everywhere finite function on the interval there exists a trigonometric series
whose sum coincides almost-everywhere with the given function (D.E. Menshov, 1941).
The expansion of functions in series is used in different areas of mathematics: in analysis — to study functions, to look for solutions of various equations containing unknown functions in the form of series, for example, by the method of indefinite coefficients (cf. Undetermined coefficients, method of), in numerical methods for the approximate calculation of the values of functions, etc.
Already the scientists of Ancient Greece had arrived at the notion of infinite sums: the sum of the terms of an infinite geometric progression with a positive ratio less than 1 can be found in their studies. As an independent concept the notion of a series entered mathematics in the 17th century. I. Newton and G. Leibniz systematically used series to solve both algebraic and differential equations. The formal theory of series was intensively developed in the 18th century and 19th century by Jacob and Johann Bernoulli, B. Taylor, C. MacLaurin, L. Euler, J. d'Alembert, J.L. Lagrange, and others. During this period both convergent and divergent series were used, though it was not completely clear whether the operations carried out on them were legitimate. The exact theory of series was created in the 19th century on the basis of the notion of a limit by C.F. Gauss, B. Bolzano, A.L. Cauchy, P.G.L. Dirichlet, N.H. Abel, K. Weierstrass, B. Riemann, and others.
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Series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Series&oldid=13797