sequences and series - limit $limlimits_{ntoinfty}left(sumlimits_{i=1}^{n}frac{1}{sqrt{i}} - 2sqrt{n}right)$

Calculate below limit

$$\lim_{n\to\infty}\left(\sum_{i=1}^{n}\frac{1}{\sqrt{i}} - 2\sqrt{n}\right)$$

Answer

As a consequence of Euler's Summation Formula, for $s > 0$, $s \neq 1$ we have

$$\sum_{j =1}^n \frac{1}{j^s} = \frac{n^{1-s}}{1-s} + \zeta(s) + O(|n^{-s}|),$$
where $\zeta$ is the Riemann zeta function.

In your situation, $s=1/2$, so
$$\sum_{j =1}^n \frac{1}{\sqrt{j}} = 2\sqrt{n} + \zeta(1/2) + O(n^{-1/2}) ,$$
and we have the limit
$$\lim_{n\to \infty} \left( \sum_{j =1}^n \frac{1}{\sqrt{j}} - 2\sqrt{n} \right) = \lim_{n\to \infty} \big( \zeta(1/2) + O(n^{-1/2}) \big) = \zeta(1/2).$$