A perhaps not so interesting class of examples where the sums in question are known to be either transcendental or explicitly computable algebraics is referenced here:
https://mathoverflow.net/a/33586
Perhaps that can direct you to more papers on the subject.
Edit: I should perhaps also mention that there are lattice based algorithms that can reconstruct minimal polynomials of algebraics with a good enough approximations. Thus, given that the degree and logarithmic height of these values are bounded by computable constants, you may take a large partial sum, use said lattice techniques, and if you don't get a match conclude that it is in fact a transcendental.
