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Convex combinations of some convergent sequences.
Convex combinations of some convergent sequences.
- 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 :
- *REAL numbers
Subjects
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