A Riemann-Type Theorem for Segmentally Alternating Series

Banakiewicz, Michał; Hanson, Bruce; Pierce, Pamela; Prus-Wiśniowski, Franciszek

doi:10.1007/s41980-018-0092-z

Cited by 3 publications

(2 citation statements)

References 2 publications

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“…Recently several results related to the Riemann rearrangement theorem and conditional convergence of series were published by several authors, see [2,10,11,13,14,16,17,[20][21][22].…”

Section: Resultsmentioning

confidence: 99%

On the $c_0$-equivalence and permutations of series

Bartoszewicz¹,

Fechner²,

Świątczak³

et al. 2020

Preprint

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Assume that a convergent series of real numbers ∞ n=1 an has the property that there exists a set A ⊆ N such that the series n∈A an is conditionally convergent. We prove that for a given arbitrary sequence (bn) of real numbers there exists a permutation σ : N → N such that σ(n) = n for every n / ∈ A and (bn) is c 0 -equivalent to a subsequence of the sequence of partial sums of the seriesMoreover, we discuss a connection between our main result with the classical Riemann series theorem.2010 Mathematics Subject Classification. 40A05, 40A35. Key words and phrases. c 0 -equivalence, center of distances, potentially convergent series, von Neumann's theorem, Riemann rearrangement theorem, hypernumber.

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“…Recently several results related to the Riemann rearrangement theorem and conditional convergence of series were published by several authors, see [2,10,11,13,14,16,17,[20][21][22].…”

Section: Resultsmentioning

confidence: 99%

On the $c_0$-equivalence and permutations of series

Bartoszewicz¹,

Fechner²,

Świątczak³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Recently, several results related to the Riemann rearrangement theorem and conditional convergence of series were published by several authors, see [2,5,9,10,[12][13][14][15][16][19][20][21].…”

Section: Corollary 22mentioning

confidence: 99%

On the $$c_0$$-equivalence and permutations of series

et al. 2021

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Assume that a convergent series of real numbers $$\sum \limits _{n=1}^\infty a_n$$ ∑ n = 1 ∞ a n has the property that there exists a set $$A\subseteq {\mathbb {N}}$$ A ⊆ N such that the series $$\sum \limits _{n \in A} a_n$$ ∑ n ∈ A a n is conditionally convergent. We prove that for a given arbitrary sequence $$(b_n)$$ ( b n ) of real numbers there exists a permutation $$\sigma :{\mathbb {N}}\rightarrow {\mathbb {N}}$$ σ : N → N such that $$\sigma (n) = n$$ σ ( n ) = n for every $$n \notin A$$ n ∉ A and $$(b_n)$$ ( b n ) is $$c_0$$ c 0 -equivalent to a subsequence of the sequence of partial sums of the series $$\sum \limits _{n=1}^\infty a_{\sigma (n)}$$ ∑ n = 1 ∞ a σ ( n ) . Moreover, we discuss a connection between our main result with the classical Riemann series theorem.

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Sets of Sums of a Series Depending on Sign Distributions

Banakiewicz

2019

Anal Math

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A Riemann-Type Theorem for Segmentally Alternating Series

Cited by 3 publications

References 2 publications

On the $c_0$-equivalence and permutations of series

On the $c_0$-equivalence and permutations of series

On the $$c_0$$-equivalence and permutations of series

Sets of Sums of a Series Depending on Sign Distributions

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