Convergence of sub-series' and sub-signed series' in terms of the asymptotic $ψ$-density
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| Format: | Preprint |
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2024
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| _version_ | 1866913461552807936 |
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| author | Heittokangas, Janne Latreuch, Zinelaabidine |
| author_facet | Heittokangas, Janne Latreuch, Zinelaabidine |
| contents | Given a non-negative real sequence $\{c_n\}_n$ such that the series $\sum_{n=1}^{\infty}c_n$ diverges, it is known that the size of an infinite subset $A\subset\mathbb{N}$ can be measured in terms of the linear density such that the sub-series $\sum_{n\in A}c_n$ either (a) converges or (b) still diverges. The purpose of this research is to study these convergence/divergence questions by measuring the size of the set $A\subset\mathbb{N}$ in a more precise way in terms of the recently introduced asymptotic $ψ$-density. The convergence of the associated sub-signed series $\sum_{n=1 }^{\infty}m_nc_n$ is also discussed, where $\{m_n\}_n$ is a real sequence with values restricted to the set $\{-1, 0, 1\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_03973 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of sub-series' and sub-signed series' in terms of the asymptotic $ψ$-density Heittokangas, Janne Latreuch, Zinelaabidine Classical Analysis and ODEs 40A05, 11B05 Given a non-negative real sequence $\{c_n\}_n$ such that the series $\sum_{n=1}^{\infty}c_n$ diverges, it is known that the size of an infinite subset $A\subset\mathbb{N}$ can be measured in terms of the linear density such that the sub-series $\sum_{n\in A}c_n$ either (a) converges or (b) still diverges. The purpose of this research is to study these convergence/divergence questions by measuring the size of the set $A\subset\mathbb{N}$ in a more precise way in terms of the recently introduced asymptotic $ψ$-density. The convergence of the associated sub-signed series $\sum_{n=1 }^{\infty}m_nc_n$ is also discussed, where $\{m_n\}_n$ is a real sequence with values restricted to the set $\{-1, 0, 1\}$. |
| title | Convergence of sub-series' and sub-signed series' in terms of the asymptotic $ψ$-density |
| topic | Classical Analysis and ODEs 40A05, 11B05 |
| url | https://arxiv.org/abs/2408.03973 |