Sum of divergent series I saw a lot of article in Math SE like Why does 1+2+3+⋯=−1/12? and S=1+10+100+100+10000+…=−1/9? How is that and lot of others. Also I saw this one of Ramanujan summation but I do not get the contradiction.
I do not want to explain how the sum of such series is calculated since I read these articles but I want an explanation of the logic of these series.


*

*Are these a contradictory results? 

*Where is the logic behind such series? 

*How come the sum of infinite positive numbers is equal to a negative one? 

*Is the problem with infinity $\infty$? 

*If someone uses this result then this someone can create a lot of absurd results ($1=0$), how to explain this please? 


I appreciate your help. Thanks.
 A: 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
A: 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
A: *

*The results are generally contradictory if you attempt to manipulate the results, and there are certain properties these results attempt to maintain, and if you mess around with them too much, you may lose certain properties, including regularity, linearity, stability, and finite re-indexing.

*The logic behind these series that produce results that generally don't make sense occur because we have developed formulas, results, and methods to find results that make sense when they can make sense.  Then someone decided "hey, this doesn't seem like a logical idea, but why don't I apply these ideas to stuff that it shouldn't work on?"  And then, well, you know... And then we were like "wow, these results agree with other results that don't make sense, so I guess if one result is right, that result must also be right, and so must the next... even if none of them should make sense at all..."

*As a wise man once said, the infinite diverging summation may do some analytical pole stuff, thus producing illogical results like a lot of positive numbers adding up to negative numbers, stuff I don't really understand.

*The problem with the infinity is the analytical pole stuff...

*The results can produce odd things, like I said, because of the way you attempt to manipulate your result and summation, I mention the properties in the first bullet.


I hope I've made a contribution and, as my teacher once said, "I hope that you leave today more comfortably confused than yesterday."
