(D) Let $S_n = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \ldots + \frac{1}{\sqrt{n}}$.For any $k \geq 1$,we have $\sqrt{k} \leq \sqrt{n}$,which im

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(D) Let $S_n = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \ldots + \frac{1}{\sqrt{n}}$.For any $k \geq 1$,we have $\sqrt{k} \leq \sqrt{n}$,which implies $\frac{1}{\sqrt{k}} \geq \frac{1}{\sqrt{n}}$.Summing this inequality from $k=1$ to $n$:$S_n = \sum_{k=1}^{n} \frac{1}{\sqrt{k}} \geq \sum_{k=1}^{n} \frac{1}{\sqrt{n}}$.$S_n \geq n \times \frac{1}{\sqrt{n}} = \sqrt{n}$.Thus,$S_n \geq \sqrt{n}$ for all $n \in N$.

Class 11 Mathematics Sequences and Series nth term of special series, Sum to n terms and Infinite number of terms

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For all $n \in N$,the sum $S_n = 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n}}$ satisfies which of the following inequalities?

AP EAMCET 2017 All AP EAMCET

A
$> n$
B
$< \sqrt{n}$
C
$\leq \sqrt{n}$
D
$\geq \sqrt{n}$

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nth term of special series, Sum to n terms and Infinite number of terms

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For all $n \in N$,the sum $S_n = 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n}}$ satisfies which of the following inequalities?

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