When studying Fourier series, the summation method that is ideal to get uniform convergence of the series to the function is Cesaro summation. See Fejer's theorem.
There are functions that are analytic, for which the Taylor series converges to the function. But there are functions for which this is not true. Using linear summation methods, like Cesaro summation you get the function back. This is a theorem of Carleman that all functions in a quasianalytic Denjoy-Carleman class can be recovered in this way, for certain linear summation method (not only Cesaro's).
You may want to look also to a nice theorem describing all linear summation methods (consistent with the usual summation) by Silverman and Toeplitz.
There are non-linear summation methods too. For example Shanks summation and its generalizations using Pade' approximants.
Notice that the way we assign a sum to a series is, to begin with, already kind of an arbitrary choice. Then, why not consider others.