Proof of an inequality: $\sqrt{n} < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}}$ [duplicate]

How do I prove the following? $$\sqrt{n} < \dfrac{1}{\sqrt{1}} + \dfrac{1}{\sqrt{2}} + \cdots + \dfrac{1}{\sqrt{n}},$$ for all $n \in\mathbb{Z}$, $n\ge 2$.

-

marked as duplicate by Srivatsan, t.b., Martin Sleziak, Asaf Karagila, Rudy the ReindeerJan 4 '12 at 21:54

Multiply both sides by $\sqrt{n}$? – user4143 Dec 6 '11 at 20:53

For any $r \in [1, n]$, we have $$\frac{1}{\sqrt{r}} \geqslant \frac{1}{\sqrt{n}}.$$ with strict inequality for $1 \leqslant r \lt n$. Adding all these $n$ inequalities, $$\sum_{r=1}^{n} \frac{1}{\sqrt{r}} \gt n \cdot \frac{1}{\sqrt{n}} = \sqrt{n}.$$

Proof using induction. The OP requested a proof using induction. I am assuming you can handle the base case $n=2$.

Now, assume that the inequality holds for some $n \geqslant 2$; we will verify the inequality for $n+1$: \begin{align*} \sum_{r=1}^{n+1} \frac{1}{\sqrt{r}} &= \sum_{r=1}^{n} \frac{1}{\sqrt{r}} + \frac{1}{\sqrt{n+1}} \\ &\gt \sqrt{n} + \frac{1}{\sqrt{n+1}} \\ &= \sqrt{n+1} + \frac{1}{\sqrt{n+1}} - (\sqrt{n+1} - \sqrt{n}) \\ &= \sqrt{n+1} + \frac{1}{\sqrt{n+1}} - \frac{1}{\sqrt{n+1} + \sqrt{n}} \\ &\gt \sqrt{n+1} , \end{align*} which is what we want to show.

Notice that out second inequality is a bit too crude. robjohn's answer shows how to get a better bound by being more careful.

-
Is there any way to prove it using mathematical induction? – geraldgreen Dec 5 '11 at 23:37
I am not sure why you would want an induction proof. Even if it were possible at all, it will definitely be only more difficult and less enlightening. – Srivatsan Dec 5 '11 at 23:39
Yes, but if one does a routine induction the proof will look harder than the nice simple argument given by Srivatsan. I can do one if you will promise not to accept it. – André Nicolas Dec 5 '11 at 23:44
I have added an induction argument, @John. – Srivatsan Dec 5 '11 at 23:45
@Srivatsan: with induction, you can get a better bound. However, for the bound requested, there is no reason to use induction. – robjohn Dec 5 '11 at 23:50

We can even do better: $$\sqrt{n}-\sqrt{n-1}=\frac{1}{\sqrt{n}+\sqrt{n-1}}<\frac{1}{2\sqrt{n-1}}\tag{1}$$ Summing, we get $$\sqrt{n}-1<\frac{1}{2}\left(\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{n-1}}\right)\tag{2}$$ So, for $n\ge2$, $$\sqrt{n}<1+\frac{1}{2}\left(\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{n-1}}\right)\tag{3}$$ which is a better bound for every $n$.

-