On Qi’s Normalized Remainder of Maclaurin Power Series Expansion of Logarithm of Secant Function

Abstract

In the study, the authors introduce Qi’s normalized remainder of the Maclaurin power series expansion of the function lnsecx=−lncosx; in view of a monotonicity rule for the ratio of two Maclaurin power series and by virtue of the logarithmic convexity of the function (2x−1)ζ(x) on (1,∞), they prove the logarithmic convexity of Qi’s normalized remainder; with the aid of a monotonicity rule for the ratio of two Maclaurin power series, the authors present the monotonic property of the ratio between two Qi’s normalized remainders.

Keywords

• Bernoulli number
• Dirichlet eta function
• Maclaurin power series expansion
• Riemann zeta function
• Stirling number of the second kind
• Qi’s normalized remainder