MathDB

1 <2\sqrt{n} - \sum 1/ \sqrt{k} <2

Source: Netherlands - Dutch MO 1987 p2

December 25, 2022

inequalitiesalgebra

Problem Statement

For

x >0

, prove that

\frac{1}{2\sqrt{x+1}}<\sqrt{x+1}-\sqrt{x}<\frac{1}{2\sqrt{x}}

and for all

n \ge 2

prove that

1 <2\sqrt{n} - \sum_{k=1}^n\frac{1}{\sqrt{k}}<2