Closed form of $\lim\limits_{n\to\infty}\left(\int_0^{n}\frac{{\rm d}k}{\sqrt{k}}-\sum_{k=1}^n\frac1{\sqrt k}\right)$

Let us represent the finite sum $\sum_{k=1}^n\frac{1}{\sqrt{k}}$ as \begin{align*}\sum_{k=1}^n\frac{1}{\sqrt{k}}&=\sum_{k=1}^n\frac{1}{\sqrt \pi}\int_0^{\infty} \frac{e^{-kx}dx}{\sqrt x}=\frac{1}{\sqrt \pi}\int_0^{\infty} e^{-x}\frac{1-e^{-(n+1)x}}{1-e^{-x}}\frac{dx}{\sqrt x}=\\ &=\frac{1}{\sqrt \pi}\int_0^{\infty} \frac{1-e^{-(n+1)x}}{\sqrt x}\left(\frac{1}{e^x-1}-\frac{e^{-x}}x+\frac{e^{-x}}x\right)dx=\\&= \frac{1}{\sqrt \pi}\int_0^{\infty} \frac{1-e^{-(n+1)x}}{\sqrt x}\left(\frac{1}{e^x-1}-\frac{e^{-x}}x\right)dx+\frac{1}{\sqrt \pi}\int_0^{\infty} \frac{e^{-x}-e^{-(n+2)x}}{x\sqrt x}dx=\\ &=\frac{1}{\sqrt \pi}\int_0^{\infty} \frac{1-e^{-(n+1)x}}{\sqrt x}\left(\frac{1}{e^x-1}-\frac{e^{-x}}x\right)dx+2\sqrt{n+2}-2. \end{align*} Since $\sqrt{n+2}-\sqrt n\to 0$ as $n\to \infty$, the limit we are looking for is given by $$L=2-\frac{1}{\sqrt \pi}\int_0^{\infty} \left(\frac{1}{e^x-1}-\frac{e^{-x}}x\right)\frac{dx}{\sqrt x}.$$ It is not difficult to relate this expression to zeta value $\zeta\left(\frac12\right)$. Indeed, the integral $$I(s)=\frac{1}{\sqrt \pi}\int_0^{\infty} x^{s-1}\left(\frac{1}{e^x-1}-\frac{e^{-x}}x\right)dx$$ converges and defines an analytic function of $s$ in the region $\Re s>0$. Furthermore for $\Re s>1$ we can break it into two separate pieces, which leads to evaluation \begin{align*}I(s)=\color{blue}{\frac{1}{\sqrt \pi}\int_0^{\infty} \frac{x^{s-1} dx}{e^x-1}}-&\color{red}{\frac{1}{\sqrt \pi}\int_0^{\infty}x^{s-2}e^{-x}dx}=\frac{\color{blue}{\Gamma(s)\zeta(s)}-\color{red}{\Gamma(s-1)}}{\sqrt\pi}=\\ &=\frac{\Gamma(s)}{\sqrt\pi}\left[\zeta(s)-\frac{1}{s-1}\right]. \end{align*} This finally gives $L=2-I\left(\frac12\right)=-\zeta\left(\frac12\right)$.

I suppose that you want to show that the sequence $u_n=2\sqrt{n}-\sum_{k=1}^n \frac{1}{\sqrt{k}}$ is convergent. We compute $v_n=u_{n+1}-u_n$. I have found $$v_n=\frac{1}{(\sqrt{n+1})(\sqrt{n}+\sqrt{n+1})^2}$$ As the series $\sum v_n$ is convergent, the sequence $u_n$ is convergent, and the limit $L$ is $L=1+\sum_{k\geq 1} v_k$.