<s>[INST] $\left\lfloor \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{1024}}\right\rfloor =?$ $$\left\lfloor \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{1024}}\right\rfloor =?$$ 
I can't find $$? \leq  \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{n}} \leq ?$$ I remeber there was a sharp bound , Am i right  ?
I am thankful if someone help me to find the bound  , or give another idea
Last question : Is it possible to find $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{1024}}$  by integral  ?
 [/INST] By the EML formula
$$ H_n^{(1/2)} = 2\sqrt{n}+\zeta\left(\tfrac{1}{2}\right)+\frac{1}{2\sqrt{n}}+O\left(\frac{1}{n^{3/2}}\right) $$
hence $H_{1024}^{(1/2)}\approx \color{red}{62}+\frac{1}{2}$. There are many questions on MSE asking for approximations of $H_n^{(1/2)}$: the Hermite-Hadamard inequality/the trapezoid method for the estimation of $\int_{1}^{1024}\frac{dx}{\sqrt{x}}$ provide straightforward answers for the task at hand.
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