A complex function is analytic if it can be represented by a Taylor Series that is valid in the neighbourhood of any chosen point in the domain of that function.
Question: How can we show the Riemann Zeta function, as represented by the series $\zeta(s)=\sum 1/n^s$ for $\Re(s)>1$ is analytic?
Can we also show the extended function $\zeta(s)=(1-2^{1-s})^{-1}\eta(s)$ for $\Re(s)>0$ is analytic?
Discussion
My challenge arises from the fact that the above representations for the Riemann zeta function are Dirichlet series, not power series which would be more amenable to comparisons to a Taylor Series.
Is it better to approach the challenge via the theorem which states that a complex function which is complex-differentiable (Cauchy-Riemann conditions) is also analytic? Would a proof to show the Dirichlet series is complex-differentiable be easier?