View source for Series

''infinite sum''

A sequence of elements (called the terms of the given series) of some [[Linear topological space|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|limit]] is defined. Here are the simplest examples of series.

==Simple series of numbers.==
A pair of sequences of complex numbers $\{a_n\}$ and $\{s_n \}$ such that
\begin{equation}\label{eq:1}
s_n = a_1+\dots+a_n, \quad n=1,2,\dots,
\end{equation}
is called a (simple) series of numbers and is denoted as follows:
\begin{equation}
a_1+a_2+\dots +a_n+\dots, \quad \sum\limits_{n=1}^{\infty}a_n
\end{equation}
or
\begin{equation}\label{eq:2}
\sum a_n.
\end{equation}
The elements of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s0846706.png" /> are called the terms of the series and the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s0846707.png" /> are called its partial sums; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s0846708.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467010.png" />-th term of the series (2), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467011.png" /> its partial sum of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467014.png" />. The series (2) is defined uniquely by each of the two sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467016.png" />: the terms of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467017.png" /> are obtained from the terms of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467018.png" /> by formula (1), and the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467019.png" /> can be recovered from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467020.png" /> by the formulas

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467021.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467022.png" /> has a finite limit

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467023.png" /></td> </tr></table>

which is called the sum of the series (2) and is written as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467024.png" /></td> </tr></table>

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|Divergent series]]).

An example of a convergent series is the sum of the terms of an infinite geometric progression

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467026.png" />. In this case its sum is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467027.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467028.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467029.png" />, (3) is an example of a divergent series.

If the series (2) is convergent, then the sequence of terms tends to zero:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467030.png" /></td> </tr></table>

The converse of this statement is not true: The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467031.png" /> of terms of the [[Harmonic series|harmonic series]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467032.png" /></td> </tr></table>

tends to zero though this series is divergent.

The series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467033.png" /> is called the remainder of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467035.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467036.png" /> of the series (2) is convergent and its sum is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467037.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467038.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467039.png" /></td> </tr></table>

If the series (2) and the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

are convergent, then the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467041.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467042.png" /> is a complex number, then the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467043.png" />, called the product of the series (2) and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467044.png" />, is also convergent, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467045.png" />.

A condition for the convergence of a series which does not use the notion of its sum is the [[Cauchy criteria|Cauchy criterion]] for the convergence of a series.

If all terms of the series (2) are real numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467046.png" />, then the series (2) is called real. In the theory of series an important role is played by real series with non-negative terms:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

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:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467048.png" /></td> </tr></table>

therefore, in this case one writes

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467049.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

with non-negative terms there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467051.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467052.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467053.png" />-th term with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467054.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467055.png" /> in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467056.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467057.png" /> is some constant), and to take the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467058.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

as a comparison series. This series is convergent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467059.png" /> and divergent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467060.png" />.

The rule below follows from the comparison test in case one takes the series (7) as a comparison series: If

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467061.png" /></td> </tr></table>

then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467063.png" /> the series (5) converges, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467065.png" /> the series (5) diverges.

The comparison test also implies the [[D'Alembert criterion (convergence of series)|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|Bertrand criterion]]; [[Gauss criterion|Gauss criterion]]; [[Lobachevskii_criterion_(for_convergence)|Lobachevskii criterion]]; [[Ermakov convergence criterion|Ermakov criterion]]; [[Kummer criterion|Kummer criterion]]; and [[Raabe criterion|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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467067.png" />. Let a series (5) be such that there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467068.png" />, defined and decreasing for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467069.png" />, the values of which on the integers coincide with the terms of the given series: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467071.png" />. Then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467072.png" /> are the partial sums and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467073.png" /> are the remainders of (5), the following estimates are valid:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467074.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467075.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467076.png" /> is some constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467077.png" />.

Thus, the series (5) converges if and only if the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467078.png" /> is convergent.

If the series (5) is divergent, then its partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467079.png" /> increase in the same way as the integrals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467080.png" />, i.e. the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467081.png" /> are asymptotically equal to the indicated integrals:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467082.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467083.png" /></td> </tr></table>

A necessary condition for the convergence of a series (5) with a decreasing sequence of terms is the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467084.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

The example of the divergent series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467085.png" /></td> </tr></table>

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|absolutely convergent series]], i.e. series (2) for which the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467086.png" /> 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467087.png" /></td> </tr></table>

The sum of a conditionally convergent series depends on the order in which its terms are written (see [[Riemann theorem|Riemann theorem]] on the rearrangement of the terms of a series): Whatever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467089.png" /> belonging to the set of real numbers completed by the infinities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467092.png" />, one can rearrange the terms of any conditionally convergent series with real numbers as its terms so that for the partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467093.png" /> of the resulting series the following equalities will hold:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467094.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467095.png" /></td> </tr></table>

is divergent, but the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467097.png" /> 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|Leibniz criterion]] for convergence is valid. Different criteria for the convergence of arbitrary series of numbers can be obtained by the [[Abel transformation|Abel transformation]] of the sums of pairwise products, for example, the [[Abel criterion|Abel criterion]]; the [[Dedekind criterion (convergence of series)|Dedekind criterion (convergence of series)]]; the [[Dirichlet criterion (convergence of series)|Dirichlet criterion (convergence of series)]]; and the [[Du Bois-Reymond criterion (convergence of series)|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467098.png" />, i.e. the products in which the sum of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s08467099.png" /> has the same value:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670100.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

and the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670101.png" /> 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:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670102.png" /></td> </tr></table>

Let the series (2), (4) and (9) be convergent and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670103.png" /></td> </tr></table>

If the series (2) and (4) are absolutely convergent, then the series (9) is also absolutely convergent and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670104.png" />. If the series (2) is absolutely convergent and the series (4) is convergent, then (9) is convergent and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670105.png" /> (Mertens' theorem). If the series (2) and (4) are conditionally convergent, then (9) may be divergent; for example, the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670106.png" /></td> </tr></table>

is conditionally convergent and the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670107.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670108.png" /></td> </tr></table>

is divergent (its terms do not tend to 0). If all three series — (2), (4) and (9) — are convergent, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670109.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670110.png" /> in which the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670111.png" /> of the indices has a fixed value:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670112.png" /></td> </tr></table>

then the product of the series (2) and (4) is defined as the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670113.png" />. This rule of multiplication is suggested by the formula for multiplication of [[Dirichlet series|Dirichlet series]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670114.png" /></td> </tr></table>

There also exist series with terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670115.png" /> numbered by all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670116.png" />. They are denoted by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670117.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

A series (10) is called convergent if the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670118.png" /></td> </tr></table>

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|multiple series]], which have terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670119.png" /> provided with multi-indices, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670120.png" /> are positive integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670121.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670122.png" />. In the theory of multiple series various types of partial sums are considered: triangular

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670123.png" /></td> </tr></table>

rectangular

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670124.png" /></td> </tr></table>

spherical

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670125.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670126.png" />, a multiple series is called a [[Double series|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:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670127.png" /></td> </tr></table>

the same is true for the values of definite integrals in which the primitives of the integrands cannot be written in elementary functions:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670128.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670129.png" /></td> </tr></table>

==Series of functions.==
A (simple) series of functions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670130.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>

is a pair of sequences of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670131.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670132.png" /> consisting of numerical functions defined on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670133.png" /> and such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670134.png" /></td> </tr></table>

As in the case of series of numbers, the elements of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670135.png" /> are called the terms of the series (11) and those of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670136.png" /> — its partial sums. The series (11) is called convergent on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670137.png" /> if for each fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670138.png" /> the following series of numbers is convergent:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670139.png" /></td> </tr></table>

Example. The series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670140.png" /></td> </tr></table>

is convergent on the entire complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670141.png" /> and the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670142.png" /></td> </tr></table>

only when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670143.png" />.

The sum of a convergent series of functions continuous, for example, on some interval is not necessarily a continuous function; for example, the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670144.png" /></td> </tr></table>

is convergent on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670145.png" />, its terms are continuous on this interval but the sum

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670146.png" /></td> </tr></table>

is discontinuous at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670147.png" />. 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|Uniformly-convergent series]]).

Series of measurable functions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670148.png" /> be a Lebesgue-measurable subset of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670149.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670150.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670151.png" /> be the Lebesgue measure and let the terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670152.png" /> of the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670153.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>

be measurable, almost-everywhere finite functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670154.png" /> taking values in the extended real line (i.e. together with real numbers they can assume the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670155.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670156.png" />). If the series (12) is convergent almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670157.png" />, then its sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670158.png" /> is also a measurable function, and by Egorov's theorem (cf. [[Egorov theorem|Egorov theorem]]), if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670159.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670160.png" /> there exists a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670161.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670162.png" /> and such that the series with as terms the restrictions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670163.png" /> of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670164.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670165.png" /> converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670166.png" /> and its sum is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670167.png" /> — the restriction of the sum (12) to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670168.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670169.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670170.png" />, be the space of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670171.png" /> with the norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670172.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670173.png" /> and with the norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670174.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670175.png" />, respectively. A series (12) is called convergent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670176.png" /> if the sequence of its partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670177.png" /> converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670178.png" />, and its limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670179.png" /> is called the sum of (12) in this space:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670180.png" /></td> </tr></table>

If a series (12) converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670181.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670182.png" /> is its sum, then there exists a subsequence of the sequence of its partial sums which converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670183.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670184.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670185.png" /> on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670186.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670187.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670188.png" /> the partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670189.png" /> of the series (12) satisfy the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670190.png" /></td> </tr></table>

if the series (12) converges almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670191.png" /> and if its sum is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670192.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670193.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>

Theorem 2. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670195.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670196.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670197.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670198.png" />, and if the sequence of partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670199.png" /> of the series (12) converges weakly to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670200.png" /> (i.e. for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670202.png" />, the following condition is satisfied:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670203.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670204.png" />. For such a series the sequence of their partial sums at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670205.png" /> increases and thus has a finite or infinite limit, which is called the value of the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670206.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670207.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670208.png" /> are uniformly bounded:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670209.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670210.png" /> is a constant, then the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670211.png" /> of the series (12) is a summable function.

Term-by-term differentiation of series. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670212.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670213.png" />-dimensional Euclidean space of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670214.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670215.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670216.png" /> be an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670217.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670218.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670219.png" /> be the generalized derivative of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670220.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670221.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670222.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670223.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670224.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670225.png" /> is fixed), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670226.png" />, and if the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670227.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670228.png" />, are convergent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670229.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670230.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670231.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670232.png" /> has a generalized derivative in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670233.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670234.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670235.png" />, i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670236.png" /></td> </tr></table>

in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670237.png" />.

Among series of functions, especially important are [[Power series|power series]]; [[Fourier series|Fourier series]]; [[Dirichlet 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:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670238.png" /></td> </tr></table>

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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670239.png" /> there exists a trigonometric series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084670/s084670240.png" /></td> </tr></table>

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|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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)  {{MR|1025126}} {{MR|0708717}} {{MR|0630899}} {{MR|0435771}} {{MR|0377444}} {{MR|0234241}} {{MR|0215962}} {{MR|0118796}} {{MR|1530727}} {{MR|0118795}} {{MR|0085462}} {{MR|0070045}} {{ZBL|0932.46001}} {{ZBL|0672.46001}} {{ZBL|0501.46001}} {{ZBL|0501.46002}} {{ZBL|0235.46001}} {{ZBL|0103.08801}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.N. Luzin,   "Theory of functions of a real variable" , Moscow  (1948)  (In Russian)  {{MR|0036819}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,   "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)  {{MR|}} {{ZBL|0307.46024}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)  {{MR|0030620}} {{ZBL|0032.05801}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.S. Bakhvalov,   "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)  {{MR|0362811}} {{ZBL|0524.65001}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L.D. Kudryavtsev,   "A course in mathematical analysis" , '''1–3''' , Moscow  (1988–1989)  (In Russian)  {{MR|1070567}} {{MR|1070566}} {{MR|1070565}} {{MR|0767983}} {{MR|0767982}} {{MR|0628614}} {{MR|0619214}} {{ZBL|0703.26001}} {{ZBL|0485.26002}} {{ZBL|0485.26001}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)  {{MR|}} {{ZBL|0397.00003}} {{ZBL|0384.00004}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  V.V. Nemytskii,   M.I. Sludskaya,   A.N. Cherkasov,   "A course of mathematical analysis" , '''1–2''' , Moscow-Leningrad  (1944)  {{MR|}} {{ZBL|}} </TD></TR></table>



====Comments====
Cf. also [[Summation methods|Summation methods]]; [[Summation of divergent series|Summation of divergent series]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.J. Bromwich,   "An introduction to the theory of infinite series" , Macmillan  (1949)  {{MR|1521495}} {{ZBL|0901.40001}} {{ZBL|0133.00801}} {{ZBL|0004.00705}}  {{ZBL|52.0208.05}}  {{ZBL|39.0306.02}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K.R. Stromberg,   "Introduction to classical real analysis" , Wadsworth  (1981)  {{MR|0604364}} {{ZBL|0454.26001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)  {{MR|0171116}} {{ZBL|0129.28002}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Knopp,   "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964)  (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)  {{MR|0028430}} {{ZBL|0124.28302}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)  pp. Chapt. X  {{MR|0933759}} {{ZBL|0628.42001}} </TD></TR></table>