# Double Series and Sums

Formalized Mathematics (2014)

• Volume: 22, Issue: 1, page 57-68
• ISSN: 1426-2630

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## Abstract

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In this paper the author constructs several properties for double series and its convergence. The notions of convergence of double sequence have already been introduced in our previous paper [18]. In section 1 we introduce double series and their convergence. Then we show the relationship between Pringsheim-type convergence and iterated convergence. In section 2 we study double series having non-negative terms. As a result, we have equality of three type sums of non-negative double sequence. In section 3 we show that if a non-negative sequence is summable, then the sequence of rearrangement of terms is summable and it has the same sums. In the last section two basic relations between double sequences and matrices are introduced.

## How to cite

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Noboru Endou. "Double Series and Sums." Formalized Mathematics 22.1 (2014): 57-68. <http://eudml.org/doc/266640>.

@article{NoboruEndou2014,
abstract = {In this paper the author constructs several properties for double series and its convergence. The notions of convergence of double sequence have already been introduced in our previous paper [18]. In section 1 we introduce double series and their convergence. Then we show the relationship between Pringsheim-type convergence and iterated convergence. In section 2 we study double series having non-negative terms. As a result, we have equality of three type sums of non-negative double sequence. In section 3 we show that if a non-negative sequence is summable, then the sequence of rearrangement of terms is summable and it has the same sums. In the last section two basic relations between double sequences and matrices are introduced.},
author = {Noboru Endou},
journal = {Formalized Mathematics},
keywords = {double series},
language = {eng},
number = {1},
pages = {57-68},
title = {Double Series and Sums},
url = {http://eudml.org/doc/266640},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Noboru Endou
TI - Double Series and Sums
JO - Formalized Mathematics
PY - 2014
VL - 22
IS - 1
SP - 57
EP - 68
AB - In this paper the author constructs several properties for double series and its convergence. The notions of convergence of double sequence have already been introduced in our previous paper [18]. In section 1 we introduce double series and their convergence. Then we show the relationship between Pringsheim-type convergence and iterated convergence. In section 2 we study double series having non-negative terms. As a result, we have equality of three type sums of non-negative double sequence. In section 3 we show that if a non-negative sequence is summable, then the sequence of rearrangement of terms is summable and it has the same sums. In the last section two basic relations between double sequences and matrices are introduced.
LA - eng
KW - double series
UR - http://eudml.org/doc/266640
ER -

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