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sum of sqrt(1-k^2/n^2) inequality

Source: VJIMC 1998 1.4-M

August 2, 2021
inequalitiesintegrationcalculusRiemann sum

Problem Statement

Prove the inequality nπ4−18n≤12+∑k=1n−11−k2n2≤nπ4\frac{n\pi}4-\frac1{\sqrt{8n}}\le\frac12+\sum_{k=1}^{n-1}\sqrt{1-\frac{k^2}{n^2}}\le\frac{n\pi}4for every integer n≥2n\ge2.
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