A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine

Abstract

In this study, by virtue of a derivative formula for the ratio of two differentiable functions and with aid of a monotonicity rule, the authors expand a logarithmic expression involving the cosine function into the Maclaurin power series in terms of specific determinants and prove a decreasing property of the ratio of two logarithmic expressions containing the cosine function.

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

“…As performed in [7] (Remark 7), using the relations (25) and (26), we can restate main results in [3,4] as follows.…”

“…In the papers [3][4][5][6], among other things, Qi introduced the functions SinR n (x) and CosR n (x) by…”

“…(1 − u) 2n−1 cos(xu) du (7) for n ∈ N and x ∈ R. We call these two quantities SinR n (x) and CosR n (x) the nth normalized tails, or the nth normalized remainders, of the Maclaurin power series expansions (1) and (2), respectively. In the papers [3][4][5][6][7], Qi and his coauthors discovered many properties of the normalized tails SinR n (x) and CosR n (x), such as their positivity, the Maclaurin power series expansions of their logarithms, the monotonicity of the ratios of logarithms of them, their concavity, and the like. For example 1.…”

“…In [3] (Theorem 2), the function ln CosR 1 (x) ln cos x = ln 1 F 2 1; 3 2 , 2; − x 2 4 ln cos x was proven to be decreasing and to map 0, π 2 onto 0, 1 6 .…”

“…32 597196800(ln 2) 2 + 259031040 ln 2 − 284891627 > 0 H(3) (0) = 96 796262400(ln 2) 3 + 518062080(ln 2) 2 + 33423360 ln 2 − 211859761 > 0 H(4) (0) = 768 398131200(ln 2) 4 + 345374720(ln 2) 3 + 33423360(ln 2) 2 − 46801281 > 0…”