Most summation methods come equipped with Tauberian theorems, which basically say that given some conditions on quickly the terms diverge, then if the method gives an answer, that answer is basically unique. Most summations methods (that have stood the test of time) are neatly arranged in a hierarchy so that if a 'weaker' method works, then all 'stronger' methods (those which can deal with greater divergence) will work and give the same answer. Hardy's book covers all this material in detail.

Another good modern source is Balser's "From divergent series to analytic differential equations", which does a great job at digesting Ecalle's theory of resurgent functions and resommability and giving it back in terms that mere mortals can understand. You might also enjoy a nice overview, by Christiane Rousseau Divergent series: past, present, future.

Most summation methods come equipped with Tauberian theorems, which basically say that given some conditions on how quickly the terms diverge, then if the method gives an answer, that answer is basically unique. Most summations methods (that have stood the test of time) are neatly arranged in a hierarchy so that if a 'weaker' method works, then all 'stronger' methods (those which can deal with greater divergence) will work and give the same answer. Hardy's book covers all this material in detail.

Another good modern source is Balser's "From divergent series to analytic differential equations", which does a great job at digesting Ecalle's theory of resurgent functions and resommability and giving it back in terms that mere mortals can understand. You might also enjoy a nice overview, by Christiane Rousseau Divergent series: past, present, future.