infinite sum
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.
A pair of sequences of complex numbers
is called a (simple) series of numbers and is denoted as follows:
or
The elements of the sequence
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
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
If the series (2) is convergent, then the sequence of terms tends to zero:
The converse of this statement is not true: The sequence
tends to zero though this series is divergent.
The series
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
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,
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
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
as a comparison series. This series is convergent for
The rule below follows from the comparison test in case one takes the series (7) as a comparison series: If
then for
The comparison test also implies the d'Alembert criterion (convergence of series) and several Criteria attributed to Cauchy (see Cauchy test). For such series there also exist the criteria of Bertrand, Gauss, Lobachevskii, Ermakov, Kummer, and Raabe (cf. Bertrand criterion; Gauss criterion; Lobachevskii criterion; Ermakov 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:
and
where
Thus, the series (5) converges if and only if the integral
If the series (5) is divergent, then its partial sums
For a series (5) whose terms form a decreasing sequence the following Cauchy condensation theorem is valid (cf. Cauchy test): 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
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
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
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
and the 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
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
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
then the product of the series (2) and (4) is defined as the series
There also exist series with terms
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
rectangular
spherical
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
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:
A (simple) series of functions
is a pair of sequences of functions
As in the case of series of numbers, the elements of the sequence
Example. The series
is convergent on the entire complex plane
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
is discontinuous at the point
Series of measurable functions. Let
be measurable, almost-everywhere finite functions on
Let
for
for
If a series (12) converges in
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
if the series (12) converges almost-everywhere on
Theorem 2. If
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
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
Theorem 4. If the terms of (12) are non-negative and if the integrals of their partial sums
where
Term-by-term differentiation of series. Let
in
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
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.
[1] | A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) MR1025126 MR0708717 MR0630899 MR0435771 MR0377444 MR0234241 MR0215962 MR0118796 MR1530727 MR0118795 MR0085462 MR0070045 Zbl 0932.46001 Zbl 0672.46001 Zbl 0501.46001 Zbl 0501.46002 Zbl 0235.46001 Zbl 0103.08801 |
[2] | N.N. Luzin, "Theory of functions of a real variable" , Moscow (1948) (In Russian) MR0036819 |
[3] | S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian) Zbl 0307.46024 |
[4] | G.H. Hardy, "Divergent series" , Clarendon Press (1949) MR0030620 Zbl 0032.05801 |
[5] | N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) MR0362811 Zbl 0524.65001 |
[6] | V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian) |
[7] | L.D. Kudryavtsev, "A course in mathematical analysis" , 1–3 , Moscow (1988–1989) (In Russian) MR1070567 MR1070566 MR1070565 MR0767983 MR0767982 MR0628614 MR0619214 Zbl 0703.26001 Zbl 0485.26002 Zbl 0485.26001 |
[8] | S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian) Zbl 0397.00003 Zbl 0384.00004 |
[9] | V.V. Nemytskii, M.I. Sludskaya, A.N. Cherkasov, "A course of mathematical analysis" , 1–2 , Moscow-Leningrad (1944) |
Cf. also Summation methods; Summation of divergent series.
[a1] | T.J. Bromwich, "An introduction to the theory of infinite series" , Macmillan (1949) MR1521495 Zbl 0901.40001 Zbl 0133.00801 Zbl 0004.00705 Zbl 52.0208.05 Zbl 39.0306.02 |
[a2] | K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) MR0604364 Zbl 0454.26001 |
[a3] | N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) MR0171116 Zbl 0129.28002 |
[a4] | K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990) MR0028430 Zbl 0124.28302 |
[a5] | A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988) pp. Chapt. X MR0933759 Zbl 0628.42001 |