Cauchy in Reverse

★ 8/10sequences Cauchy-Schwarz Telescoping inequality

Problem

Let $x_1, x_2, \dots, x_n > 0$ . Prove that

\frac{1}{x_1} + \frac{2}{x_1+x_2} + \cdots + \frac{n}{x_1+x_2+\cdots+x_n} < 2\left(\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}\right).

Answer

Solution

Difficulty	8/10
Topics	sequences, Cauchy-Schwarz, Telescoping, inequality

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