In the book I am reading they write that for Hurwitz zeta function, $\zeta(x,s)=\sum_{n=0}^{\infty} \frac{1}{(x+n)^s}$, the next sum in the RHS converges for $\Re(s)>-1$, and I don't see how exactly?!

$$\zeta(x,s)-(\zeta(s)-sx\zeta(s+1)= x^{-s} + \sum_{n=1}^{\infty} n^{-s}[(1+x/n)^{-s} - (1-x/n)]$$

I think the sum converges for $\Re (s+1) > 1$ which means $\Re s >0$ and not $>-1$.

The reference is Andrews' et al Special Functions red book, page 17. http://books.google.co.il/books?id=kGshpCa3eYwC&printsec=frontcover#v=onepage&q&f=false