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.

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
