Sum of divergent series

L. Euler explained his assumptions about infinite series - convergent or divergent - with the following idea (just paraphrasing, don't have the article at hand, but you can look at the Euler-archives the treatize "De series divergentibus"): The evaluation of an infinite series is different from a finite sum. But always when we want to assign a value for such a series we should do it in the sense, that it is the result of an infinitely applied arithmetic operation - so that the geometric series (to which we meanwhile assign a value) occurs as result of the infinite formal long-division $s(x) = {1 \over 1-x } \to s(x) = 1 + x + x^2 + ... $ and then insert the value for $x$ in the finite rational formula.

Possibly this is meant in a sense, that similarly we can discuss infinite periodic continued fractions as representations of finite expressions like $\sqrt{1+x}$ and others. It is "compatible" somehow to an axiom, that we require for number theory that we can have a closed-form representation for general infinitely repeated (symbolic) algebraic operation. (in the german translation of E247 this occurs in §11 and §12)

From this, I think, for instance Euler-summation and other manipulations on infinite (convergent and divergent) series by L. Euler can be nicely understood.

[update] The Euler-archives seem to have moved to MAA; the original links, for instance //www.eulerarchive.com/ is taken over by some completely unrelated commercials. A seemingly valid link to Ed Sandifer's column "How Euler did it", however only accessible via internal MAA-access is this (but I think via webarchive.org one can still access the former existent openly available pages)

[update 2]: here is a currently valid link to Ed Sandifer's article

I have a new idea.

The sum of the natural numbers is $$ S_n = \sum_{k=1}^n k. $$ We define the function $$ G_n(\epsilon) = \sum_{k=1}^n k \exp(-k\epsilon). $$ Abel sum is $$ S_A = \lim_{\epsilon \to 0+} \left( \lim_{n \to \infty} G_n(\epsilon) \right). $$ Unfortunately, it diverges.

Then we define a new function $$ H_n(\epsilon) = \sum_{k=1}^n k \exp(-k\epsilon) \cos(k\epsilon). $$ The function is damped and oscillating. The damped oscillation sum is $$ S_H = \lim_{\epsilon \to 0+} \left( \lim_{n \to \infty} H_n(\epsilon) \right). $$ Surprisingly, it converges on -1/12.

We can confirm the result by the numerical computation. Please input the following formula to the page of Wolfram Alpha.

lim sum k exp(-kx)cos(kx),k=1 to infty,x to 0+

Or click the following URL with the above formula, please.

https://www.wolframalpha.com/input/?i=lim+sum+k+exp%28-kx%29cos%28kx%29%2Ck%3D1+to+infty%2Cx+to+0%2B

We can find the paper by searching the following keywords.

Zeta function regularization of the sum of all natural numbers by damped oscillation summation method