Time for me to eat some crow.
I am looking at
Several weeks ago we had a problem from a member that posted a summation that seemed impossible. As I recall the consensus was that the problem was an example of how not to use the theorems about summing infinite series. (If someone knows the link please let me know. I can't find it.) Well, I've run into it myself now. I am quoting from "String Theory and M-Theory" by Becker, Becker, and Schwarz, page 50.
I've never been what you would call "good" at analytic continuation. How the heck can you do this?... (sum on the right hand side) is divergent and needs to be regularized. This can be achieved using -function regularization. First, one considers the general sum
which is defined for any complex number s. For Re(s) > 1, this sum converges to the Riemann zeta function . This zeta function has a unique analytic continuation to s = -1, where it takes the value ....