I am trying to obtain the meromorphic continuation of the Riemann zeta function $\zeta(s)$ for all $s \in \mathbb{C}$ with $Re(s) > 0$. Using the Abel's summation formula I've obtained that $$ \sum_{n \leq x}\frac{1}{n^s}=\frac{\lfloor x \rfloor}{x} + s\int_1^x \frac{\lfloor t \rfloor}{t^{s+1}}dt. $$ Letting $x \longrightarrow \infty$ we have $$ \zeta(s)= \frac{s}{s-1} - s\int_1^{\infty} \frac{\{t\}}{t^{s+1}}dt $$ for all $s$ with $Re(s) > 1$.
How can I continue after this point?
The fact that $$s \mapsto \int_1^{\infty} \frac{\{t\}}{t^{s+1}}dt$$ is a holomorphic function on $\{\operatorname{Re}(s) > 0 \}$ follows from the dominated convergence theorem: it tells you that you may interchange the order of differentiation and integration.