Decreasing ratio between two normalized remainders of Maclaurin series expansion of exponential function

Tao Zhang; Feng Qi; Tao Zhang; Feng Qi

doi:10.3934/math.2025663

AIMS Mathematics

2025, Volume 10, Issue 6: 14739-14756. doi: 10.3934/math.2025663

Previous Article Next Article

Review

Decreasing ratio between two normalized remainders of Maclaurin series expansion of exponential function

Tao Zhang ¹,
Feng Qi ^{2,3,4
,
,}

1.
Key Laboratory of Infinite-dimensional Hamiltonian System and its Algorithm Application, Ministry of Education, Inner Mongolia Normal University, Hohhot 010022, China
2.
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, China
3.
Retired researcher, 17709 Sabal Court, Dallas, TX 75252-8024, USA
4.
School of Mathematics and Physics, Hulunbuir University, Inner Mongolia 021008, China

Received: 27 February 2025 Revised: 13 June 2025 Accepted: 24 June 2025 Published: 26 June 2025
MSC : Primary 41A58; Secondary 26A48, 33B10

In the study, the authors concisely review the normalized remainders of the Maclaurin series and, by establishing an inequality for specific Maclaurin series, show that the ratio between two normalized remainders of the Maclaurin series of the exponential function is decreasing on the whole real axis. This decreasing property confirms a guess in Remark 5 of the paper "F. Qi, Absolute monotonicity of normalized tail of power series expansion of exponential function, Mathematics, 12 (2024), 2859, available online at https://doi.org/10.3390/math12182859".
- decreasing ratio,
- normalized remainder,
- exponential function,
- Maclaurin series,
- inequality for series
Citation: Tao Zhang, Feng Qi. Decreasing ratio between two normalized remainders of Maclaurin series expansion of exponential function[J]. AIMS Mathematics, 2025, 10(6): 14739-14756. doi: 10.3934/math.2025663

Related Papers:

Abstract

In the study, the authors concisely review the normalized remainders of the Maclaurin series and, by establishing an inequality for specific Maclaurin series, show that the ratio between two normalized remainders of the Maclaurin series of the exponential function is decreasing on the whole real axis. This decreasing property confirms a guess in Remark 5 of the paper "F. Qi, Absolute monotonicity of normalized tail of power series expansion of exponential function, Mathematics, 12 (2024), 2859, available online at https://doi.org/10.3390/math12182859".

References

[1]	H. Alzer, J. L. Brenner, O. G. Ruehr, Inequalities for the tails of some elementary series, J. Math. Anal. Appl., 179 (1993), 500–506. https://doi.org/10.1006/jmaa.1993.1364 doi: 10.1006/jmaa.1993.1364
[2]	Z. H. Bao, R. P. Agarwal, F. Qi, W. S. Du, Some properties on normalized tails of Maclaurin power series expansion of exponential function, Symmetry, 16 (2024), 989. https://doi.org/10.3390/sym16080989 doi: 10.3390/sym16080989
[3]	M. Dubourdieu, Sur un théorème de M. S. Bernstein relatif à la transformation de Laplace-Stieltjes, Compos. Math., 7 (1940), 96–111.
[4]	A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of integral transforms: Bateman manuscript project, New York: McGraw-Hill Book Company, 1954.
[5]	I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, 8 Eds., Amsterdam: Academic Press, 2015.
[6]	Y. F. Li, F. Qi, A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine, Open Math., 21 (2023), 20230159. https://doi.org/10.1515/math-2023-0159 doi: 10.1515/math-2023-0159
[7]	Y. W. Li, F. Qi, W. S. Du, Two forms for Maclaurin power series expansion of logarithmic expression involving tangent function, Symmetry, 15 (2023), 1686. https://doi.org/10.3390/sym15091686 doi: 10.3390/sym15091686
[8]	X. L. Liu, H. X. Long, F. Qi, A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing sine, Mathematics, 11 (2023), 3107. https://doi.org/10.3390/math11143107 doi: 10.3390/math11143107
[9]	X. L. Liu, F. Qi, Monotonicity results of ratio between two normalized remainders of Maclaurin series expansion for square of tangent function, Math. Slovaca, 75 (2025), 699–705. https://doi.org/10.1515/ms-2025-0051 doi: 10.1515/ms-2025-0051
[10]	D. S. Mitrinović, J. E. Pečarić, A. M. Fink, Classical and new inequalities in analysis, Dordrecht: Springer, 1993. https://doi.org/10.1007/978-94-017-1043-5
[11]	W. J. Pei, B. N. Guo, Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine, Open Math., 22 (2024), 20240095. https://doi.org/10.1515/math-2024-0095 doi: 10.1515/math-2024-0095
[12]	F. Qi, Absolute monotonicity of normalized tail of power series expansion of exponential function, Mathematics, 12 (2024), 2859. https://doi.org/10.3390/math12182859 doi: 10.3390/math12182859
[13]	F. Qi, Series and connections among central factorial numbers, Stirling numbers, inverse of Vandermonde matrix, and normalized remainders of Maclaurin series expansions, Mathematics, 13 (2025), 223. https://doi.org/10.3390/math13020223 doi: 10.3390/math13020223
[14]	F. Qi, R. P. Agarwal, D. Lim, Decreasing property of ratio of two logarithmic expressions involving tangent function, Math. Comp. Model. Dyn., 31 (2025), 2449322. https://doi.org/10.1080/13873954.2024.2449322 doi: 10.1080/13873954.2024.2449322
[15]	R. L. Schilling, R. Song, Z. Vondraček, Bernstein functions: theory and applications, Berlin: De Gruyter, 2012. https://doi.org/10.1515/9783110269338
[16]	N. M. Temme, Special functions: an introduction to the classical functions of mathematical physics, New York: John Wiley & Sons, Inc., 1996. https://doi.org/10.1002/9781118032572
[17]	F. Wang, F. Qi, Absolute monotonicity of four functions involving the second kind of complete elliptic integrals, J. Math. Inequal., in press.
[18]	F. Wang, F. Qi, Power series expansion and decreasing property related to normalized remainders of power series expansion of sine, Filomat, 38 (2024), 10447–10462. https://doi.org/10.2298/FIL2429447W doi: 10.2298/FIL2429447W
[19]	D. V. Widder, The Laplace transform (PMS-6), Princeton: Princeton University Press, 1946.
[20]	A. Xu, Some identities connecting Stirling numbers, central factorial numbers and higher-order Bernoulli polynomials, AIMS Mathematics, 10 (2025), 3197–3206. https://doi.org/10.3934/math.2025148 doi: 10.3934/math.2025148
[21]	H. C. Zhang, B. N. Guo, W. S. Du, On Qi's normalized remainder of Maclaurin power series expansion of logarithm of secant function, Axioms, 13 (2024), 860. https://doi.org/10.3390/axioms13120860
[22]	J. Zhang, F. Qi, Some properties of normalized remainders of the Maclaurin expansion for a function originating from an integral representation of the reciprocal of the gamma function, Math. Inequal. Appl., 28 (2025), 343–354. https://doi.org/10.7153/mia-2025-28-23 doi: 10.7153/mia-2025-28-23
[23]	G. Z. Zhang, F. Qi, On convexity and power series expansion for logarithm of normalized tail of power series expansion for square of tangent, J. Math. Inequal., 18 (2024), 937–952. https://doi.org/10.7153/jmi-2024-18-51 doi: 10.7153/jmi-2024-18-51
[24]	G. Z. Zhang, Z. H. Yang, F. Qi, On normalized tails of series expansion of generating function of Bernoulli numbers, Proc. Amer. Math. Soc., 153 (2025), 131–141. https://doi.org/10.1090/proc/16877 doi: 10.1090/proc/16877

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)