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Convex combinations of some convergent sequences.

Convex combinations of some convergent sequences.

Authors :
Stević, Stevo
Source :
Mathematical Methods in the Applied Sciences. Sep2024, p1. 14p.
Publication Year :
2024

Abstract

We consider the convex combinations cnα:=(1−α)an+αbn,n∈ℕ,α∈[0,1]$$ {c}_n^{\alpha}:= \left(1-\alpha \right){a}_n+\alpha {b}_n,n\in \mathrm{\mathbb{N}},\alpha \in \left[0,1\right] $$, of a pair of sequences of real numbers (an)n∈ℕ$$ {\left({a}_n\right)}_{n\in \mathrm{\mathbb{N}}} $$ and (bn)n∈ℕ$$ {\left({b}_n\right)}_{n\in \mathrm{\mathbb{N}}} $$ such that an≤bn,n∈ℕ$$ {a}_n\le {b}_n,n\in \mathrm{\mathbb{N}} $$, converging to ln2$$ \ln 2 $$, and study the location of the limit inside the intervals [an,bn]$$ \left[{a}_n,{b}_n\right] $$, for every n∈ℕ$$ n\in \mathrm{\mathbb{N}} $$ or for sufficiently large n$$ n $$. We also investigate the same problem for the case of two corresponding sequences converging to ln3$$ \ln 3 $$. Among other results, we prove some, a bit, unexpected ones. Namely, for each α∈[0,1]$$ \alpha \in \left[0,1\right] $$, we determine the exact index n0∈ℕ$$ {n}_0\in \mathrm{\mathbb{N}} $$ at which the sequence cnα$$ {c}_n^{\alpha } $$ changes the monotonicity, and we also determine the type of the monotonicity. A number of interesting remarks are also presented. [ABSTRACT FROM AUTHOR]

Subjects

Subjects :
*REAL numbers

Details

Language :
English
ISSN :
01704214
Database :
Academic Search Index
Journal :
Mathematical Methods in the Applied Sciences
Publication Type :
Academic Journal
Accession number :
179320266
Full Text :
https://doi.org/10.1002/mma.10463