Convergence Question: This is related to the Dirichlet eta function. Does 
$$\int_1^\infty \frac{dx}{x^z}$$
converge for $Re(z)>1$? Just wondering. If so, then does 
$$\int_1^2 \frac{dx}{x^z}+\int_3^4 \frac{dx}{x^z}+\cdots+\int_n^{n+1} \frac{dx}{x^z}+\int_{n+2}^{n+3} \frac{dx}{x^z}+\cdots$$
Converge as well?

Yes, and yes. $\big($Why exactly did you think they wouldn't ?$\big)$.
