As x goes to infinity the sum
may be evaluated
Sorry, I don't follow what you're trying to say. The sum may be approximated by the integral, but they're not equal, and the integral isn't equal to so I'm not quite sure what you're trying to show...could you explain a bit more please?
This sum is a Riemann sum of this integral.
For x going to infinity
OK I see what you're saying now. I think everything's fine as long as we can show that the approximation of the sum by the integral at the beginning is valid; that is, we need to show
as , where means as . I've thought about this for a while; is it just a standard kind of integration result or can you prove it?