<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>{rn}n</italic>∈𝕅 be a strictly increasing sequence of nonnegative real numbers satisfying the asymptotic formula <italic>rn ~ αβn</italic>, where <italic>α, β</italic> are real numbers with <italic>α ></italic> 0 and <italic>β ></italic> 1. In this note we prove some limits that connect this sequence to the number <italic>e</italic>. We also establish some asymptotic formulae and limits for the counting function of this sequence. All of the results are applied to some well-known sequences in mathematics.</abstract>

Let {rn}n∈𝕅 be a strictly increasing sequence of nonnegative real numbers satisfying the asymptotic formula rn ~ αβn, where α, β are real numbers with α > 0 and β > 1. In this note we prove some limits that connect this sequence to the number e. We also establish some asymptotic formulae and limits for the counting function of this sequence. All of the results are applied to some well-known sequences in mathematics.

Notes on a General Sequence

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"Notes on a General Sequence"}

Let {rn}n∈𝕅 be a strictly increasing sequence of nonnegative real numbers satisfying the asymptotic formula rn ~ αβn, where α, β are real numbers with α > 0...

Notes on a General Sequence

Published Online: May 08, 2020

Page range: 193 - 202

Received: Jan 28, 2020

Accepted: Apr 18, 2020

DOI: https://doi.org/10.2478/amsil-2020-0006

Keywords
general sequence, asymptotic formulae, limit behavior, the number

© 2020 Reza Farhadian et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Notes on a General Sequence

Published Online: May 08, 2020

Page range: 193 - 202

Received: Jan 28, 2020

Accepted: Apr 18, 2020

DOI: https://doi.org/10.2478/amsil-2020-0006

Keywordsgeneral sequence, asymptotic formulae, limit behavior, the number

© 2020 Reza Farhadian et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
general sequence, asymptotic formulae, limit behavior, the number