Chiph, I believe that the series should be:
Here is my solution:
There is one online but it's just a special case of this one. It's for x = 1. They do reference the paper where it was derived and I'm sure they did it for arbitrary x. Maybe that's what you saw. Thanks though.
I think the series could be derived with a straight application of Euler summation: Euler summation - Wikipedia, the free encyclopedia