Elegant proofs for properties of normalized remainders of Maclaurin power series expansion of exponential function

Abstract In the paper, by establishing an integral representation of a specific Maclaurin power series and in light of two monotonicity rules, the authors present very elegant proofs for several basic properties, including the positivity, (absolute) monotonicity, logarithmic convexity, and inequalities, of the normalized remainders of the Maclaurin power series expansion of the exponential function.