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★ 8/10Absolute Value sequences Estimation inequality

Problem

Let $n > 2$ and let $a_1, \dots, a_n \in (0, 1]$ . With $A_k = \dfrac{a_1 + \cdots + a_k}{k}$ , prove that

\left|\sum_{k=1}^{n} a_k - \sum_{k=1}^{n} A_k\right| < \frac{n-1}{2}.

Answer

Solution

Difficulty	8/10
Topics	Absolute Value, sequences, Estimation, inequality

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