Power series expansion and decreasing property related to normalized remainders of power series expansion of sine

Abstract

In the paper, with the aid of a derivative formula for the ratio of two differentiable functions, in view of a monotonicity rule for the ratio of two differentiable functions, in terms of the Hessenberg determinants, the authors present the Maclaurin power series expansion of the logarithm of the normalized remainder of the Maclaurin power series expansion of the sine function and demonstrate the decreasing property of the ratio of two logarithms of two normalized remainders of the Maclaurin power series expansion of the sine function.