Bounding $\sqrt{1} + \sqrt{2} + \cdots + \sqrt{n}$

Problem

Prove that for every positive integer $n$ ,

\frac{2n}{3}\sqrt{n} \;<\; \sum_{k=1}^{n} \sqrt{k} \;<\; \frac{4n+3}{6}\sqrt{n}.

Answer

Solution

Difficulty	7/10
Topics	Induction, inequality

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