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A series
 
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/d/d033/d033980/d0339801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\sum _ {m , n = 1 } ^  \infty  u _ {mn} ,
 +
$$
  
the terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d0339802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d0339803.png" /> of which form a double sequence of numbers. The finite sums
+
the terms $  u _ {mn} $,
 +
$  m , n = 1 , 2 \dots $
 +
of which form a double sequence of numbers. The finite sums
  
<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/d/d033/d033980/d0339804.png" /></td> </tr></table>
+
$$
 +
S _ {mn}  = \sum _ {i = 1 } ^ { m }
 +
\sum _ {j = 1 } ^ { n }  u _ {ij}  $$
  
are said to be the partial sums of the double series (1) or its rectangular partial sums. They also form a double sequence. If this sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d0339805.png" /> has a finite [[Double limit|double limit]]
+
are said to be the partial sums of the double series (1) or its rectangular partial sums. They also form a double sequence. If this sequence $  \{ S _ {mn} \} $
 +
has a finite [[Double limit|double 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/d/d033/d033980/d0339806.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
= \lim\limits _ {m , n \rightarrow \infty }  S _ {mn} ,
 +
$$
  
the series (1) is said to be convergent, and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d0339807.png" /> is said to be its sum:
+
the series (1) is said to be convergent, and the number $  S $
 +
is said to be its 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/d/d033/d033980/d0339808.png" /></td> </tr></table>
+
$$
 +
= \sum _ {m , n = 1 } ^  \infty  u _ {mn} .
 +
$$
  
 
If there is no finite limit (2), the series (1) is said to be divergent. Double series have many of the properties of ordinary (single) series. For instance, if the double series
 
If there is no finite limit (2), the series (1) is said to be divergent. Double series have many of the properties of ordinary (single) series. For instance, if the double 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/d/d033/d033980/d0339809.png" /></td> </tr></table>
+
$$
 +
\sum _ {m , n = 1 } ^  \infty  a _ {mn} ,\ \
 +
\sum _ {m , n = 1 } ^  \infty  b _ {mn}  $$
  
converge, then, for any numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398011.png" />, the double series
+
converge, then, for any numbers $  \lambda $
 +
and $  \mu $,  
 +
the double 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/d/d033/d033980/d03398012.png" /></td> </tr></table>
+
$$
 +
\sum _ {m , n = 1 } ^  \infty  ( \lambda a _ {mn} + \mu b _ {mn} )
 +
$$
  
 
also converges and
 
also converges 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/d/d033/d033980/d03398013.png" /></td> </tr></table>
+
$$
 +
\sum _ {m , n = 1 } ^  \infty  ( \lambda a _ {mn} + \mu b _ {mn} )
 +
= \lambda \sum _ {m , n = 1 } ^  \infty  a _ {mn} + \mu
 +
\sum _ {m , n = 1 } ^  \infty  b _ {mn} .
 +
$$
  
 
If a double series converges, then
 
If a double series converges, 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/d/d033/d033980/d03398014.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {m , n \rightarrow \infty }  u _ {mn}  = 0
 +
$$
  
(a necessary condition for convergence of the series (1)). For the double series (1) to converge it is necessary and sufficient that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398015.png" /> there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398016.png" /> such that
+
(a necessary condition for convergence of the series (1)). For the double series (1) to converge it is necessary and sufficient that for any $  \epsilon > 0 $
 +
there exists a number $  N _  \epsilon  $
 +
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/d/d033/d033980/d03398017.png" /></td> </tr></table>
+
$$
 +
| S _ {m + k , n + l }  - S _ {mn} |  < \epsilon ,
 +
$$
  
provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398019.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398021.png" /> are arbitrary non-negative integers. If all terms of the series (1) are non-negative, the sequence of its partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398022.png" /> always has a finite or an infinite limit, and
+
provided that $  m > N _  \epsilon  $,  
 +
$  n > N _  \epsilon  $,  
 +
and $  k $
 +
and $  l $
 +
are arbitrary non-negative integers. If all terms of the series (1) are non-negative, the sequence of its partial sums $  S _ {mn} $
 +
always has a finite or an infinite limit, 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/d/d033/d033980/d03398023.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {m , n \rightarrow \infty }  S _ {mn}  = \
 +
\sup _ {m , n = 1, 2 ,\dots }  S _ {mn} .
 +
$$
  
The specific properties of a double series are due to the presence of double-indexed terms. If the double series (1) converges and if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398024.png" /> the series
+
The specific properties of a double series are due to the presence of double-indexed terms. If the double series (1) converges and if for all $  n = 1 , 2 \dots $
 +
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/d/d033/d033980/d03398025.png" /></td> </tr></table>
+
$$
 +
\sum _ {m = 1 } ^  \infty  u _ {mn} ,
 +
$$
  
 
converge as well, then the repeated series
 
converge as well, then the repeated 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/d/d033/d033980/d03398026.png" /></td> </tr></table>
+
$$
 +
\sum _ {n = 1 } ^  \infty 
 +
\left ( \sum _ {m = 1 } ^  \infty  u _ {mn} \right )
 +
$$
  
 
also converges, and its sum is equal to the sum of the given series.
 
also converges, and its sum is equal to the sum of the given series.
Line 51: Line 105:
 
A double series is said to converge absolutely if the series
 
A double series is said to converge absolutely 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/d/d033/d033980/d03398027.png" /></td> </tr></table>
+
$$
 +
\sum _ {m , n = 1 } ^  \infty  | u _ {mn} |
 +
$$
  
 
converges. If a double series is absolutely convergent, it is also convergent; moreover, any series obtained by rearrangement of its terms also converges, and the sum of any such arbitrary series is the same as the sum of the initial series.
 
converges. If a double series is absolutely convergent, it is also convergent; moreover, any series obtained by rearrangement of its terms also converges, and the sum of any such arbitrary series is the same as the sum of the initial series.
Line 57: Line 113:
 
A double series whose terms are functions displays many properties of ordinary series of functions and many concepts are common to both, including the concept of uniform convergence, the Cauchy criterion of uniform convergence of a series or the Weierstrass criterion of uniform convergence. Nevertheless, many theorems valid for ordinary series cannot be directly applied to double series. Thus, the direct analogue of the [[Abel theorem|Abel theorem]] for power series
 
A double series whose terms are functions displays many properties of ordinary series of functions and many concepts are common to both, including the concept of uniform convergence, the Cauchy criterion of uniform convergence of a series or the Weierstrass criterion of uniform convergence. Nevertheless, many theorems valid for ordinary series cannot be directly applied to double series. Thus, the direct analogue of the [[Abel theorem|Abel theorem]] for 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/d/d033/d033980/d03398028.png" /></td> </tr></table>
+
$$
 +
\sum _ {n= 0 } ^  \infty  a _ {n} x  ^ {n}
 +
$$
  
 
does not apply to double power series, i.e. to series of the type
 
does not apply to double power series, i.e. to series of the type
  
<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/d/d033/d033980/d03398029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\sum _ {m,n= 0 } ^  \infty  c _ {mn} x  ^ {m} y  ^ {n} .
 +
$$
  
 
There exist, for example, double series (3) which converge at two points in the plane only: The series (3) with coefficients
 
There exist, for example, double series (3) which converge at two points in the plane only: The series (3) with coefficients
  
<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/d/d033/d033980/d03398030.png" /></td> </tr></table>
+
$$
 +
c _ {0n}  = c _ {n0}  = - c _ {1n}  = - c _ {n1}  = n! ,\ \
 +
n= 1, 2 \dots
 +
$$
  
<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/d/d033/d033980/d03398031.png" /></td> </tr></table>
+
$$
 +
c _ {mn}  = 0, m, n\geq  2 ,
 +
$$
  
converges only at the two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398033.png" />.
+
converges only at the two points $  ( 0, 0) $
 +
and $  ( 1, 1) $.
  
 
Besides the definition (2) for a double series (1) there also exist other definitions of its convergence and its sum, which are also connected with the double indexation of its terms. For instance, let
 
Besides the definition (2) for a double series (1) there also exist other definitions of its convergence and its sum, which are also connected with the double indexation of its terms. For instance, 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/d/d033/d033980/d03398034.png" /></td> </tr></table>
+
$$
 +
S _ {p}  = \sum _ {m+ n \leq  p } u _ {mn} ,\  p= 1, 2 ,\dots
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398035.png" /> is said to be a triangular partial sum of the double series (1)); the double series (1) will be convergent if the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398036.png" /> is convergent; its limit
+
( $  S _ {p} $
 +
is said to be a triangular partial sum of the double series (1)); the double series (1) will be convergent if the sequence $  \{ S _ {p} \} $
 +
is convergent; its 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/d/d033/d033980/d03398037.png" /></td> </tr></table>
+
$$
 +
= \lim\limits _ {p \rightarrow \infty }  S _ {p} ,
 +
$$
  
 
is known as the triangular sum of the series (1). If one puts
 
is known as the triangular sum of the series (1). If one puts
  
<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/d/d033/d033980/d03398038.png" /></td> </tr></table>
+
$$
 +
S _ {r}  = \sum _ {m  ^ {2} + n  ^ {2} \leq  r  ^ {2} }
 +
u _ {mn} ,\  r> 0
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398039.png" /> is said to be a circular partial sum), then the double series (1) is said to be convergent if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398040.png" /> of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398041.png" /> has a limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398042.png" />, and this limit
+
( $  S _ {r} $
 +
is said to be a circular partial sum), then the double series (1) is said to be convergent if the function $  S _ {r} $
 +
of the parameter $  r $
 +
has a limit as $  r \rightarrow + \infty $,  
 +
and this 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/d/d033/d033980/d03398043.png" /></td> </tr></table>
+
$$
 +
= \lim\limits _ {r \rightarrow + \infty } S _ {r}  $$
  
 
is said to be the circular limit of the series (1).
 
is said to be the circular limit of the series (1).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398044.png" /> denote an arbitrary finite set of index pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398045.png" /> and put
+
Let $  {\mathcal N} $
 +
denote an arbitrary finite set of index pairs $  ( m, n) $
 +
and put
  
<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/d/d033/d033980/d03398046.png" /></td> </tr></table>
+
$$
 +
S _  {\mathcal N}    = \sum _ {( m, n) \in {\mathcal N} } u _ {mn} .
 +
$$
  
Then the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398047.png" /> is said to be the sum of the series (1) if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398048.png" /> there exists a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398049.png" /> of pairs of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398050.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398051.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398052.png" /> is valid. If such a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033980/d03398053.png" /> exists, the series (1) is convergent.
+
Then the number $  S $
 +
is said to be the sum of the series (1) if for any $  \epsilon > 0 $
 +
there exists a finite set $  {\mathcal N} _  \epsilon  $
 +
of pairs of indices $  ( m, n) $
 +
such that for any $  {\mathcal N} \supset {\mathcal N} _  \epsilon  $
 +
the inequality $  | S - S _  {\mathcal N}  | < \epsilon $
 +
is valid. If such a number $  S $
 +
exists, the series (1) is convergent.
  
 
The definitions of convergence of a series (1) listed above are not mutually equivalent. However, if the terms of the double series are non-negative, convergence in one of the above senses entails convergence in all other senses as well, and the values of the sums of (1) in all cases then coincide. Different summation methods exist for double series.
 
The definitions of convergence of a series (1) listed above are not mutually equivalent. However, if the terms of the double series are non-negative, convergence in one of the above senses entails convergence in all other senses as well, and the values of the sums of (1) in all cases then coincide. Different summation methods exist for double series.

Latest revision as of 19:36, 5 June 2020


A series

$$ \tag{1 } \sum _ {m , n = 1 } ^ \infty u _ {mn} , $$

the terms $ u _ {mn} $, $ m , n = 1 , 2 \dots $ of which form a double sequence of numbers. The finite sums

$$ S _ {mn} = \sum _ {i = 1 } ^ { m } \sum _ {j = 1 } ^ { n } u _ {ij} $$

are said to be the partial sums of the double series (1) or its rectangular partial sums. They also form a double sequence. If this sequence $ \{ S _ {mn} \} $ has a finite double limit

$$ \tag{2 } S = \lim\limits _ {m , n \rightarrow \infty } S _ {mn} , $$

the series (1) is said to be convergent, and the number $ S $ is said to be its sum:

$$ S = \sum _ {m , n = 1 } ^ \infty u _ {mn} . $$

If there is no finite limit (2), the series (1) is said to be divergent. Double series have many of the properties of ordinary (single) series. For instance, if the double series

$$ \sum _ {m , n = 1 } ^ \infty a _ {mn} ,\ \ \sum _ {m , n = 1 } ^ \infty b _ {mn} $$

converge, then, for any numbers $ \lambda $ and $ \mu $, the double series

$$ \sum _ {m , n = 1 } ^ \infty ( \lambda a _ {mn} + \mu b _ {mn} ) $$

also converges and

$$ \sum _ {m , n = 1 } ^ \infty ( \lambda a _ {mn} + \mu b _ {mn} ) = \lambda \sum _ {m , n = 1 } ^ \infty a _ {mn} + \mu \sum _ {m , n = 1 } ^ \infty b _ {mn} . $$

If a double series converges, then

$$ \lim\limits _ {m , n \rightarrow \infty } u _ {mn} = 0 $$

(a necessary condition for convergence of the series (1)). For the double series (1) to converge it is necessary and sufficient that for any $ \epsilon > 0 $ there exists a number $ N _ \epsilon $ such that

$$ | S _ {m + k , n + l } - S _ {mn} | < \epsilon , $$

provided that $ m > N _ \epsilon $, $ n > N _ \epsilon $, and $ k $ and $ l $ are arbitrary non-negative integers. If all terms of the series (1) are non-negative, the sequence of its partial sums $ S _ {mn} $ always has a finite or an infinite limit, and

$$ \lim\limits _ {m , n \rightarrow \infty } S _ {mn} = \ \sup _ {m , n = 1, 2 ,\dots } S _ {mn} . $$

The specific properties of a double series are due to the presence of double-indexed terms. If the double series (1) converges and if for all $ n = 1 , 2 \dots $ the series

$$ \sum _ {m = 1 } ^ \infty u _ {mn} , $$

converge as well, then the repeated series

$$ \sum _ {n = 1 } ^ \infty \left ( \sum _ {m = 1 } ^ \infty u _ {mn} \right ) $$

also converges, and its sum is equal to the sum of the given series.

A double series is said to converge absolutely if the series

$$ \sum _ {m , n = 1 } ^ \infty | u _ {mn} | $$

converges. If a double series is absolutely convergent, it is also convergent; moreover, any series obtained by rearrangement of its terms also converges, and the sum of any such arbitrary series is the same as the sum of the initial series.

A double series whose terms are functions displays many properties of ordinary series of functions and many concepts are common to both, including the concept of uniform convergence, the Cauchy criterion of uniform convergence of a series or the Weierstrass criterion of uniform convergence. Nevertheless, many theorems valid for ordinary series cannot be directly applied to double series. Thus, the direct analogue of the Abel theorem for power series

$$ \sum _ {n= 0 } ^ \infty a _ {n} x ^ {n} $$

does not apply to double power series, i.e. to series of the type

$$ \tag{3 } \sum _ {m,n= 0 } ^ \infty c _ {mn} x ^ {m} y ^ {n} . $$

There exist, for example, double series (3) which converge at two points in the plane only: The series (3) with coefficients

$$ c _ {0n} = c _ {n0} = - c _ {1n} = - c _ {n1} = n! ,\ \ n= 1, 2 \dots $$

$$ c _ {mn} = 0, m, n\geq 2 , $$

converges only at the two points $ ( 0, 0) $ and $ ( 1, 1) $.

Besides the definition (2) for a double series (1) there also exist other definitions of its convergence and its sum, which are also connected with the double indexation of its terms. For instance, let

$$ S _ {p} = \sum _ {m+ n \leq p } u _ {mn} ,\ p= 1, 2 ,\dots $$

( $ S _ {p} $ is said to be a triangular partial sum of the double series (1)); the double series (1) will be convergent if the sequence $ \{ S _ {p} \} $ is convergent; its limit

$$ S = \lim\limits _ {p \rightarrow \infty } S _ {p} , $$

is known as the triangular sum of the series (1). If one puts

$$ S _ {r} = \sum _ {m ^ {2} + n ^ {2} \leq r ^ {2} } u _ {mn} ,\ r> 0 $$

( $ S _ {r} $ is said to be a circular partial sum), then the double series (1) is said to be convergent if the function $ S _ {r} $ of the parameter $ r $ has a limit as $ r \rightarrow + \infty $, and this limit

$$ S = \lim\limits _ {r \rightarrow + \infty } S _ {r} $$

is said to be the circular limit of the series (1).

Let $ {\mathcal N} $ denote an arbitrary finite set of index pairs $ ( m, n) $ and put

$$ S _ {\mathcal N} = \sum _ {( m, n) \in {\mathcal N} } u _ {mn} . $$

Then the number $ S $ is said to be the sum of the series (1) if for any $ \epsilon > 0 $ there exists a finite set $ {\mathcal N} _ \epsilon $ of pairs of indices $ ( m, n) $ such that for any $ {\mathcal N} \supset {\mathcal N} _ \epsilon $ the inequality $ | S - S _ {\mathcal N} | < \epsilon $ is valid. If such a number $ S $ exists, the series (1) is convergent.

The definitions of convergence of a series (1) listed above are not mutually equivalent. However, if the terms of the double series are non-negative, convergence in one of the above senses entails convergence in all other senses as well, and the values of the sums of (1) in all cases then coincide. Different summation methods exist for double series.

The concept of a double series can be generalized to series whose terms are not numbers but, for example, elements of a normed linear space.

References

[1] A. Pringsheim, "Elementare Theorie der unendlichen Doppelreihen" Münchener Sitzungsber. der Math. , 27 (1897) pp. 101–153
[2] A. Pringsheim, "Zur Theorie der zweifach unendlichen Zahlenfolgen" Math. Ann. , 53 (1900) pp. 289–321
[3] A. Pringsheim, "Vorlesungen über Zahlen- und Funktionenlehre" , 2 , Teubner (1932)
[4] T.J. Bromwich, "An introduction to the theory of infinite series" , Macmillan (1947)
[5] G.S. Salekhov, "Computation of series" , Moscow (1955) (In Russian)
[6] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6
How to Cite This Entry:
Double series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Double_series&oldid=46774
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article