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x_n <\1/\sqrt{n! H_n}

Source: 2011 Saudi Arabia Pre-TST November 4.3

January 1, 2022
algebrainequalities

Problem Statement

Let n≥2n \ge 2 be a positive integer and let xnx_n be a positive real root to the equation x(x+1)...(x+n)=1x(x+1)...(x + n) = 1. Prove that xn<1n!Hnx_n <\frac{1}{\sqrt{n! H_n}} where Hn=1+12+...+1nH_n = 1+\frac12+...+\frac{1}{n}.
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