Can every divergent series be regularized?

In the most general sense, a summation is a partial function from the set of summand sequences to $\mathbb R$ (or $\mathbb C$). This sounds like we could assign more or less arbitrary values and if we want we really can. However, certain properties of summations are preferred to hold, such as

• Regularity that is, our summation method should be an extension of the standard -convergent-sequence-of-partial-sums method
• Linearity that is, if we define $\sum a_n$ and $\sum b_n$ then we also define $\sum(ca_n+b_n)$ and have $\sum(ca_n+b_n)=c\sum a_n+\sum b_n$
• Stability $\sum a_n$ is defined if and only if $\sum a_{n+1}$ is defined and we have $\sum a_n=a_2+\sum a_{n+1}$

To repeat: not all summation methods (not even all methods in practical use) obaey all three criteria. But if we concentrate on methods obeying all three then indeed we often get that certain (classically) divergent series are always assigned the same value under any summation method. For example, $\sum x^n=\frac1{1-x}$ follows for all $x\ne 1$ where we define the sum by, merely playing around with stability and linearity.

So how high can we try? We can use Zorn's lemma to find a maximal regular, linear, stable summation method. But will "maximal" imply "total", i.e., that all series become summable? And will the summation thus obtained be well-defined? Unfortunately, the answer to both is no. This can already be exemplified with $\sum 1$, which has do be a solution of $\sum 1 = 1+\sum 1$ per statbility. (Then again, you have have read that regularization can assign $1+1+1+\ldots =-\frac12$; apparently those methods are not linear or nopt stable ...)