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:
or
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:
and
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
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
,
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.
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
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.
Historical remarks.
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.