I'm looking for more references for the following limit representation of the Riemann zeta function which I found on Wolfram's web site. Thanks in advance.
$$\zeta (s)=\lim_{n\to \infty } \, \left(\sum _{k=1}^n k^{-s}-\frac{n^{1-s}-1}{1-s}\right)-\frac{1}{1-s},\quad\Re(s)>0.$$
$$n^{-s} - \int_n^{n+1} x^{-s}dx = \int_n^{n+1} \int_n^x s t^{-s-1}dtdx = \mathcal{O}(n^{-s-1})$$ So that
$$\sum_{n=1}^\infty (n^{-s}-\int_n^{n+1} x^{-s}dx) = \lim_{N \to \infty} \sum_{n=1}^N n^{-s}-\int_1^{N+1} x^{-s}dx=\lim_{N \to \infty} \sum_{n=1}^N n^{-s}-\frac{1-(N+1)^{1-s}}{s-1}$$ converges and is analytic for $Re(s) > 0$.
Clearly for $Re(s) > 1$ : $\lim_{N \to \infty} \sum_{n=1}^N n^{-s}-\frac{1-(N+1)^{1-s}}{s-1} = \zeta(s)-\frac{1}{s-1}$ and by analytic continuation this stays true for $Re(s) > 0$.
