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Question Number 160694 by naka3546 last updated on 04/Dec/21

$$Provethat$$ $$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\dots +\frac{1}{\sqrt{n}}<2\sqrt{n}$$

Answered by mindispower last updated on 04/Dec/21

$$\frac{1}{\sqrt{1+x}}=\frac{2}{2\sqrt{1+x}}<\frac{2}{\sqrt{x}+\sqrt{x+1}}=2(\sqrt{1+x}-\sqrt{x})$$ $$\underset{k=0}{\sum ^{n-1}}\frac{1}{\sqrt{1+k}}<2\underset{k=0}{\sum ^{n-1}}(\sqrt{k+1}-\sqrt{k})$$ $$1+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}<2\sqrt{n}$$

Commented bynaka3546 last updated on 04/Dec/21

$$Thankyou,sir.$$

Commented bymindispower last updated on 05/Dec/21

$$withepleasurgodblessYou$$

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