There are many methods for assigning a value to a series that diverges, e.g. zeta function regularization, Abel summation, Cesaro summation, etc. From all of the examples I've found, two methods either give the same result or one of them doesn't work. For example, both zeta function regularization, Ramanujun summation, and a method of Euler assign -1/12 to 1 + 2 + 3 + 4 + … while Abel summation can't assign a value. My question is if there is an example of a series that different summation methods assign different values to or is it the case that any two summation methods must agree on divergent series (that they can assign a value to). Here I am assuming that both summation methods assign the correct value to convergent series and are linear. I am guessing we need stronger conditions since it seems that the space of convergent series is not dense, in some sense, in the space of all series.
EDIT: I was able to pick up a copy of Hardy's "Divergent Series." It's a really neat book but I have yet to be able to find in it an example of a divergent series that gets assigned two different values by two different linear and consistent summation methods. He does show how a method being linear forces specific series to have a unique value. Surely the issue of whether two general summation methods (with reasonable conditions that they satisfy) can disagree on a certain series must come up somewhere in the literature.
Best Answer
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.