If we agree to define $S=\lim_{n\to \infty}\sum_0^na_n$,then, only convergent series can have meaningful $S$. But if we decide to define $S$, to be the Caesaro mean, now we can also assign a real number to $S$ for some divergent series as well. My question is such relaxation of the definition frown upon in the mathematical community? The reason I ask is I have read several blog posts where authors are adamant that only convergent series are summable and only summmability in the ordinary sense is useful. In those articles, they usually point out the Ramanujan sum as an example of getting nonsensical answers. They especially have huge problems with getting results by analytical continuation of Rieman zeta function. To me Caesaro summation and Abelian summation are natural extensions of the summation in ordinary sense. Where do you draw the line?
P.S. I am very new to this topic and mathematics in general so please bear with me for my naive question.
Mathematicians don't have any inherent problem with Cesaro sums, regularizations, etc.; they're well-defined and useful. Cesaro sums pop up in Fourier analysis, for example, and Tauberian theorems in number theory have at least the same spirit as the sort of regularization that comes up in dealing with these divergent sums.
The problems are with treating these operations on infinite sequences as the same as the ordinary sum (i.e., the limit of partial sums) and blithely applying the same calculus to them. There's a meaningful and useful way to define a function on some infinite sequences that takes the value $−1/12$ at $(a_n) = (1, 2, 3, \dots)$, to take the inevitable example, but it's absolutely not true in any sense that the sum $1 + 2 + 3 + \cdots$ converges to $-1/12$.