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polynomial by sum (-x)^{2k}/(2k)! >=0

Source: 1974 Hungary - Kürschák Competition p3

October 15, 2022
algebrapolynomialinequalities

Problem Statement

Let pk(x)=1−x+x22!−x33!+...+(−x)2k(2k)!p_k(x) = 1 -x + \frac{x^2}{2! } - \frac{x^3}{3!}+ ... + \frac{(-x)^{2k}}{(2k)!} Show that it is non-negative for all real xx and all positive integers kk.
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