# Double Series and Sums

Formalized Mathematics (2014)

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

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topNoboru 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 -

## References

top- [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
- [2] Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.
- [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
- [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
- [5] Grzegorz Bancerek. Representation theorem for stacks. Formalized Mathematics, 19(4): 241-250, 2011. doi:10.2478/v10037-011-0033-2.[Crossref] Zbl1276.14019
- [6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
- [7] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
- [8] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
- [9] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.
- [10] Czesław Bylinski. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241-245, 1996.
- [11] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
- [12] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
- [13] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
- [14] Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.
- [15] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
- [16] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
- [17] Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167-175, 2008. doi:10.2478/v10037-008-0023-1.[Crossref] Zbl1321.46022
- [18] Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Double sequences and limits. Formalized Mathematics, 21(3):163-170, 2013. doi:10.2478/forma-2013-0018.[Crossref] Zbl1298.40003
- [19] Fuguo Ge and Xiquan Liang. On the partial product of series and related basic inequalities. Formalized Mathematics, 13(3):413-416, 2005.
- [20] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
- [21] Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181-187, 2005.
- [22] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.
- [23] Gilbert Lee. Weighted and labeled graphs. Formalized Mathematics, 13(2):279-293, 2005.
- [24] Konrad Raczkowski and Andrzej Nedzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.
- [25] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
- [26] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.
- [27] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- [28] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
- [29] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
- [30] Bo Zhang and Yatsuka Nakamura. The definition of finite sequences and matrices of probability, and addition of matrices of real elements. Formalized Mathematics, 14(3): 101-108, 2006. doi:10.2478/v10037-006-0012-1.[Crossref]

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