A Square-Root Partial-Sum Bound

Problem

Let $n \in \mathbb{N}^*$ , and let $a_1, a_2, \ldots, a_n$ be positive reals with partial sums $S_k = a_1 + \cdots + a_k$ and $S_n = 1$ . Prove that

1 \leqslant \sum_{k=1}^n \frac{a_k}{\sqrt{S_k}} < 2.

Answer

Solution

Difficulty	8/10
Topics	sequences, Telescoping, inequality

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