sequences and series - Estimating $sum n^{-1/2}$

Could someone please explain me how does one obtain the following estimate:
$$\sum_{n \leq X} n^{-1/2} = \frac12 X^{1/2} + c + O(X^{-1/2}),$$
where $c$ is some constant.

Thank you very much!

PS As pointed out in the comments, $1/2$ in front of $X^{1/2}$ is a typo... I would like an answer with the correct coefficient here.

Answer

You can use the Euler McLaurin formula which gives us the estimate

$$2 \sqrt{n} + K + \frac{1}{2\sqrt{n}} + \mathcal{O}(n^{-3/2})$$

It can be shown by other means that $K = \zeta(\frac{1}{2})$.