Some new facts about group 𝒢 generated by the family of convergent permutations

“…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

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.

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

View full text Add to dashboard Cite

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.

show abstract

Some new facts about group 𝒢 generated by the family of convergent permutations

Cited by 3 publications

References 4 publications

New properties of the families of convergent and divergent permutations - Part I

New properties of the families of convergent and divergent permutations - Part I

On the $c_0$-equivalence and permutations of series

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

Contact Info

Product

Resources

About