Word Count : 255

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(i) पहले भाग के लिए, हमें यह दिखाना है कि [ 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}} > 2\sqrt{n+1} - 2, \quad \forall n \in \mathbb{N}. ] हम मान सकते हैं कि (f(k) = \frac{1}{\sqrt{k}} > 2(\sqrt{k+1} - \sqrt{k})). इसका कारण यह है कि [ 2(\sqrt{k+1} - \sqrt{k}) = \frac{2((k+1)-k)}{\sqrt{k+1} + \sqrt{k}} = \frac{2}{\sqrt{k+1} + \sqrt{k}} < \frac{1}{\sqrt{k}}, ] क्योंकि _______ _________ _________ _______ ____ __________ ____ ______ __________.
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