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Anyone who can solve it or give me an idea on how to try to do it myself?

$$1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + ... + \frac{1}{\sqrt{n}}\geq \sqrt{n}, \;\;\;n \in \mathbb{N^*}$$

G Tony Jacobs
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Insomniac
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  • Maybe try induction. – kingW3 Nov 20 '17 at 17:36
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    This question looks very familiar. Second, the inequality should be $\ge$. – Math Lover Nov 20 '17 at 17:39
  • Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. – Fly by Night Nov 20 '17 at 17:39
  • It might help to know that $\sqrt{n+1} \le \sqrt n + \frac {1}{2\sqrt n}$ – Doug M Nov 20 '17 at 17:39
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    Shouldn't the sign be the other way around? – Stefan4024 Nov 20 '17 at 17:39
  • Many ideas there, and the inequality is wrong way. https://math.stackexchange.com/questions/2149448/other-idea-to-show-an-inequality-dfrac1-sqrt-1-dfrac1-sqrt-2-dfrac1/2149480#2149480 – zwim Nov 20 '17 at 17:42
  • yes its >= i m gonna edit – Insomniac Nov 20 '17 at 17:45

4 Answers4

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The inequality should be the other way around. To solve it multiply both sides by $\sqrt{n}$ and note that the RHS is $n$ and on the LHS you have $n$ terms each greater than $1$.

Stefan4024
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1

If we multiply both sides by $\sqrt n$, we obtain,:

$\sqrt{n}+\sqrt{\frac{n}2}+\cdots + 1\ge n$.

Each term on the left is at least $1$, and there are $n$ of them, so the sum is greater than or equal to $n$, as desired.

G Tony Jacobs
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we have $ 1<\sqrt n, \sqrt 2 < \sqrt n ... \implies \frac{1}{\sqrt 1}+\frac{1}{\sqrt 2} ...+\frac{1}{\sqrt n} \geq \frac{1}{\sqrt n}*n = \sqrt n$

Magneto
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The denominators of the terms on the left are increasing, so the smallest term on the left is $\frac1{\sqrt{n}}$. Thus

$$\frac1{1}+\frac1{\sqrt{2}}+\cdots +\frac1{\sqrt{n}}$$ $$\geq \frac1{\sqrt{n}} + \frac1{\sqrt{n}} + \cdots + \frac1{\sqrt{n}}$$ $$= n\cdot\frac1{\sqrt{n}} = \sqrt{n}$$

MPW
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