Double series

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