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2

Part of 1987 Dutch Mathematical Olympiad

Problems(1)

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

Source: Netherlands - Dutch MO 1987 p2

12/25/2022
For x>0x >0 , prove that 12x+1<x+1−x<12x\frac{1}{2\sqrt{x+1}}<\sqrt{x+1}-\sqrt{x}<\frac{1}{2\sqrt{x}} and for all n≥2n \ge 2 prove that 1<2n−∑k=1n1k<21 <2\sqrt{n} - \sum_{k=1}^n\frac{1}{\sqrt{k}}<2
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