I am interested in how one might generally evaluate, or estimate
$$G(x)=\sum_{n=1}^{\infty}f(n)x^n-\int_{0}^{\infty}f(t)x^tdt$$ as $x\to1^-$, and for a continuous $f$, and such that the integral and the power series both converge in some suitable region.
One would expect that the limit (I am only really interested in what happens when $x$ approaches 1 or some other root of unity) would generally exist, but perhaps there are some cases where the difference blows up. Any ideas on how this function $G(x)$ behaves asymptotically would be very useful.