\begin{array}{rcl} \displaystyle\sum_{1\leq n\leq x}\frac{1}{\sqrt{n}}&=&\displaystyle\frac{[x]}{\sqrt{x}}+\frac{1}{2}\int\limits_1^x\frac{[t]}{t^{3/2}}\,dt\\
&=&\displaystyle\frac{x+\mathcal{O}(1)}{\sqrt{x}}+\frac{1}{2}\int\limits_1^x\frac{t-\{t\}}{t^{3/2}}\,dt\\
&=&\displaystyle\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\\
&=&\displaystyle 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)\\
&=&\displaystyle 2\sqrt{x}+c+\mathcal{O}\left(\frac{1}{\sqrt{x}}\right),