Continuation of the Zeta Function

Question

I already showed that für $\sigma >1$, $$\zeta (s) = \frac{1}{s-1} + \frac{1}{2} + \sum_{j=1; 2\mid j+1}^{k-1}\left( \prod_{i=0}^{j-1}(s+i) \right) b_{j+1}(0) - \left( \prod_{j=0}^{k-1}(s+j)\right) \int_1^{\infty } \frac{b_k(x)}{x^{s+k}},$$ where the $b_i$ are some kind of Bernoulli polynomials. Now I have to prove that this gives an analytic continuation for the Zeta Function to the half-plane $\sigma >-k+1$. Do I just have to prove that then the integral converges? If so, how can I do this? If not, what else do I have to show?