The Best Constant Is $\tfrac23$

★ 7/10sequences Estimation Telescoping inequality

Problem

Consider the inequality

1 + \sqrt2 + \cdots + \sqrt n < C\,(n+1)^{3/2} \qquad (n \in \mathbb{N}^*).

Prove that it holds for all $n$ when $C = \dfrac23$ , and fails for some $n$ whenever $C < \dfrac23$ .

Answer

Solution

Difficulty	7/10
Topics	sequences, Estimation, Telescoping, inequality

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