Decreasing ratio between two normalized remainders of Maclaurin series expansion of exponential function

In the study, the authors concisely review the normalized remainders of the Maclaurin series and, by establishing an inequality for specific Maclaurin series, show that the ratio between two normalized remainders of the Maclaurin series of the exponential function is decreasing on the whole real axis. This decreasing property confirms a guess in Remark 5 of the paper 'F. Qi, Absolute monotonicity of normalized tail of power series expansion of exponential function, Mathematics, 12 (2024), 2859, available online at https://doi.org/10.3390/math12182859'.