Word Count : 353

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We are given the inequality: $$ 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}} \geq \sqrt{2(n-1)}, \quad \text{for } n \in \mathbb{N},\ n > 1. $$ We are to prove it manually. --- ### Approach: Use the integral comparison method. Let us denote: $$ S_n = \sum_{k=1}^{n} \frac{1}{\sqrt{k}}. $$ We aim to lower-bound this sum and show __________ _____ ______ ________ ____ _______ ____ _________ _________.
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