Question: Does $$\sum\_{i=1}^\infty \frac{1}{(a+i)^b}$$ converge for $b>1$? What is the name of this series?

Answer: 
This might helps: when $\Re s>1,\ \Re q>0$, the [Hurwitz zeta function](http://en.wikipedia.org/wiki/Hurwitz_zeta_function) is defined by
$$\zeta(s,q)=\sum\_{n=0}^\infty(n+q)^{-s}.$$
This sequence is absolutely convergent for the given $s,q$, and its domain can be extended to $\Bbb C\setminus\{1\}$ by analytic continuation. In particular,
$$\zeta(s,1)=\zeta(s).$$
Almost from definition of Hurwitz Zeta function $$\sum\_{i=1}^\infty \frac{1}{(a+i)^b}=\zeta (b,a+1)$$ which is absolutely convergent.