There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences.

Still, in my view there is fundamental difference between divergent integrals or series that diverge to infinity, monotonously growing as opposed to those which have infinite parts of opposite signs that cancel each other or can be averaged so to arrive at finite values. I would call them strongly divergent and weakly divergent. If we apply an absolute value to a function under a weakly divergent integral, it becomes strongly divergent. The concept is similar to absolute and non-absolute convergence.

Examples of integrals that "diverge" but can be averaged (weakly divergent):

$$\int_0^\infty\sin x dx=1$$ (Cesaro average)

$$\int_{-1}^1 \frac1xdx=0$$ (Cauchy mean value)

$$\int_0^\infty \left(x-\frac2{x^3}\right)dx=0$$

(applying area-preserving operator $\mathcal{L}_t[t f(t)](x)$ to one term gives the other with opposite sign, so they cancel each other)

Examples of strongly-divergent integrals:

$$\int_{-1}^1 \frac1{x^2}dx=2\pi\delta(0)-2$$

Using Hadamard finite part can be regularized to $-2$.

$$\int_0^\infty 1 dx=\pi\delta(0)$$

Can be obtained from the previous one using the area-preserving operator, regularizes to $0$.

$$\int_0^1 \frac1x dx$$

Using Ramanujan regularization of its Riemann sum (and other methods), can be regularized to $\gamma$.

$$\sum_{k=0}^\infty k$$

(can be regularized to $-\frac1{12}$ using Ramanujan's, Dirichlet's or Zeta regularization)

Thus, all summation methods can be divided into two very distinct parts: those that somehow average the result, attempting to cancel the infinite parts and those that do not make such attempts, but extract the finite part, dropping the infinite part.

Was such classification of summation methods ever been proposed?

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    $\begingroup$ all summation methods can be divided into two very distinct parts --- I don't know much about summability theory, but I would guess that this might be an oversimplification. Does this hold for the various methods discussed in Classical and Modern Methods in Summability by Johann Boos (2000) or those discussed in the mathoverflow question Ideal characterization of almost convergence or those making use of Banach limits? $\endgroup$ – Dave L Renfro May 3 at 15:46
  • $\begingroup$ Your title does not match the question in the post. The answer to the title question is that an integral "divergent but with finite average” is called "Cesaro summable". But the post is asking about classifying summation methods, not about classifying integrals. Which is it? $\endgroup$ – Conifold May 3 at 23:28
  • $\begingroup$ @Conifold Abel summation is also based on averaging, the same is true for taking Cauchy main value. Possibly, also for Borel. $\endgroup$ – Anixx May 4 at 6:15
  • $\begingroup$ So are you looking for a catchall name for integrals that can be "averaged" in any vague sense? Or is it still about classifying summation methods? $\endgroup$ – Conifold May 4 at 6:29

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