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Question Number 192661 by Mingma last updated on 24/May/23

Commented by Mingma last updated on 24/May/23

Prove that

Commented by Mingma last updated on 25/May/23

Perfect ��

Answered by deleteduser1 last updated on 24/May/23

(√1)+(√2)+...+(√(2n−1))≤(2n−1)(√(((2n−1)(2n))/(2(2n−1))))=(2n−1)(√n) ⇒(1/( (√n)))((√1)+(√2)+...+(√(2n−1)))≤2n−1

$$\sqrt{\mathrm{1}}+\sqrt{\mathrm{2}}+...+\sqrt{\mathrm{2}{n}−\mathrm{1}}\leqslant\left(\mathrm{2}{n}−\mathrm{1}\right)\sqrt{\frac{\left(\mathrm{2}{n}−\mathrm{1}\right)\left(\mathrm{2}{n}\right)}{\mathrm{2}\left(\mathrm{2}{n}−\mathrm{1}\right)}}=\left(\mathrm{2}{n}−\mathrm{1}\right)\sqrt{{n}} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\:\sqrt{{n}}}\left(\sqrt{\mathrm{1}}+\sqrt{\mathrm{2}}+...+\sqrt{\mathrm{2}{n}−\mathrm{1}}\right)\leqslant\mathrm{2}{n}−\mathrm{1} \\ $$

Commented by Mingma last updated on 25/May/23

Perfect ��

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