Power series expansion, decreasing property, and concavity related to logarithm of normalized tail of power series expansion of cosine

Abstract

In this paper, in view of a determinantal formula for higher order derivatives of the ratio of two differentiable functions, we expand the logarithm of the normalized tail of the power series expansion of the cosine function into a Maclaurin power series expansion whose coefficients are expressed in terms of specific Hessenberg determinants, present the decreasing property and concavity of the normalized tail of the Maclaurin power series expansion of the cosine function, deduce a new determinantal expression of the Bernoulli numbers, and verify the decreasing property for the ratio of the logarithms of the first two normalized tails of the Maclaurin power series expansion of the cosine function.

Keywords

• maclaurin power series expansion
• normalized tail
• cosine
• ratio
• derivative formula
• hessenberg determinant
• decreasing property
• concavity
• bernoulli number
• logarithm
• determinantal expression