On normalized tails of series expansion of generating function of Bernoulli numbers

Abstract

In the paper, the authors present the positivity and decreasing property of the normalized tails of the series expansion of the generating function of the classical Bernoulli numbers and prove the increasing property of the ratio between two normalized tails of the series expansion of the generating function of the classical Bernoulli numbers by showing the increasing property of the ratio between two Bernoulli polynomials.

Monotonicity Results of Ratios between Normalized Tails of Maclaurin Power Series Expansions of Sine and Cosine

“…Comparing the above-mentioned results obtained in [2] (Corollaries 1.3 and 1.4) by Koumandos with previous results on normalized tails and their derivations in [6,8] and Remark 7 in [7], we believe that this question and its answers should be interesting, important, significant, and useful in mathematics.…”

“…Combining the positivity of the function f n (x) defined in (8) for n ≥ 2 on (0, ∞) with the equalities in ( 12) and (13), we deduce that the function F n (x) is positive for n ≥ 1 on (0, ∞).…”

“…In Ref. [8], based on the generating function in (27) with t = 0, the authors introduced the normalized tails…”

“…In the electronic preprint at the site https://doi.org/10.48550/arxiv.2405.05280 (accessed on 6 May 2024), Yang and Qi gave an alternative proof of [8] (Proposition 1) about the monotonicity results of the ratio…”