Hello
Mathematica have a build-in command SumConvergence,but gives conditions for the sum to be convergent or not,
and nothing else. If can't find conditions then returns input.
SumConvergence[(-1)^n*Sin[1/n], n]
(* True*)
SumConvergence[(-1)^n*(2 + (-1)^n)/(3 + (-1)^n), n]
(* False *)
Regards,MI