Prove $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\dots+\frac{1}{\sqrt{n}} > 2\:(\sqrt{n+1} − 1)$ Basically, I'm trying to prove (by induction) that:
$$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\dots+\frac{1}{\sqrt{n}} > 2\:(\sqrt{n+1} − 1)$$
I know to begin, we should use a base case.  In this case, I'll use $1$.  So we have:
$$1 > 2\:(1+1-1) = 1>-2$$
Which works out.
My problem is the next step.  What comes after this?  Thanks!
 A: Let $f(x) = \frac{1}{\sqrt{x}}$. This is a decreasing function on $\mathbb{R}^+_*$, hence
$$ f(k) > \int_k^{k+1} f(x) \, \mathrm{d}x. $$
Summing for $k=1, \dots, n$ give
$$ f(1) + f(2) + \dots + f(n) > \int_1^{n+1} f(x) \, \mathrm{d}x,$$
which is exactly the desired inequality.
A: Hint(for induction):
$$2(\sqrt{n+1}-1)+\dfrac{1}{\sqrt{n+1}}>2(\sqrt{n+2}-1)\iff \\
2n+3>2\sqrt{(n+2)(n+1)}\iff \\
(2n+3)^2>4(n+2)(n+1)=(2n+3)^2-1\cdots$$
A: Mean Value Theorem can also be used,
Let $\displaystyle f(x)=\sqrt{x}$
$\displaystyle f'(x)=\frac{1}{2}\frac{1}{\sqrt{x}}$
Using mean value theorem we have:
$\displaystyle \frac{f(n+1)-f(n)}{(n+1)-n}=f'(c)$ for  some $c\in(n,n+1)$
$\displaystyle \Rightarrow \frac{\sqrt{n+1}-\sqrt{n}}{1}=\frac{1}{2}\frac{1}{\sqrt{c}}$....(1)
$\displaystyle \frac{1}{\sqrt{n+1}}<\frac{1}{\sqrt{c}}<\frac{1}{\sqrt{n}}$
Using the above ineq. in $(1)$ we have,
$\displaystyle \frac{1}{2\sqrt{n+1}}<\sqrt{n+1}-\sqrt{n}<\frac{1}{2\sqrt{n}}$
Adding the left part of the inequality we have,$\displaystyle\sum_{k=2}^{n}\frac{1}{2\sqrt{k}}<\sum_{k=2}^{n}(\sqrt{k}-\sqrt{k-1})=\sqrt{n}-1$
$\Rightarrow \displaystyle\sum_{k=2}^{n}\frac{1}{\sqrt{k}}<2\sum_{k=2}^{n}(\sqrt{k}-\sqrt{k-1})=2(\sqrt{n}-1)$
$\Rightarrow \displaystyle1+\sum_{k=2}^{n}\frac{1}{\sqrt{k}}<1+2\sum_{k=2}^{n}(\sqrt{k}-\sqrt{k-1})=2\sqrt{n}-2+1=2\sqrt{n}-1$
$\Rightarrow \displaystyle\sum_{k=1}^{n}\frac{1}{\sqrt{k}}<2\sqrt{n}-1$
Similarly adding the right side of the inequality we have,
$\displaystyle\sum_{k=1}^{n}\frac{1}{2\sqrt{k}}>\sum_{k=1}^{n}(\sqrt{k+1}-\sqrt{k})=\sqrt{n+1}-1$
$\Rightarrow \displaystyle\sum_{k=1}^{n}\frac{1}{\sqrt{k}}>2(\sqrt{n+1}-1)$
This completes the proof.
$\displaystyle 2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1.$
This is a much better proof than proving by induction(Ofcourse if someone knows elementary calculus).
A: We can prove that for $x \geqslant 1$ 
 \begin{align*}\sum_{1\leqslant n\leqslant x}\frac{1}{\sqrt{n}}&\overset{*}=\frac{[x]}{\sqrt{x}}+\frac{1}{2}\int\limits_1^x\frac{[t]}{t^{3/2}}\,dt\\ &=\frac{x+\mathcal{O}(1)}{\sqrt{x}}+\frac{1}{2}\int\limits_1^x\frac{t-\{t\}}{t^{3/2}}\,dt\\ &=\sqrt{x}+\mathcal{O}\left(\frac{1}{\sqrt{x}}\right)+\frac{1}{2}\int\limits_1^x\frac{dt}{\sqrt{t}}-\frac{1}{2}\int\limits_1^{\infty}\frac{\{t\}}{t^{3/2}}\,dt+\frac{1}{2}\int\limits_x^{\infty}\frac{\{t\}}{t^{3/2}}\,dt\\ &=2\sqrt{x}-1-\frac{1}{2}\int\limits_1^{\infty}\frac{\{t\}}{t^{3/2}}\,dt+\mathcal{O}\left(\frac{1}{\sqrt{x}}\right)\\ &=2\sqrt{x}+c+\mathcal{O}\left(\frac{1}{\sqrt{x}}\right),\end{align*}
where $\displaystyle c\overset{\text{def}}{=}-1-\frac{1}{2}\int\limits_1^{\infty}\frac{\{t\}}{t^{3/2}}\,dt $ is some absolute real constant.
$*$ - we used here Abel's summation formula for $ a_n=1 $ and $ \phi(n)=\dfrac{1}{\sqrt{n}} $
$[\,\,]$ denotes floor function and $ \{\,\,\} $ fractional part function
