I apologise for the inefficient typesetting of the question, this is my first time on the website. I was thinking about two series that are defined as below:
S1 = {-2, -3, -4, -5, -6 ... -∞} to negative infinity
S2 = {+6, +7, +8 +9, +10 ... ∞} to positive infinity.
If one were to take the sum of two series, there seems to be 2 different values for the some. One is a finite value, you can observe that in the first series given after the value of "-6" every value of same magnitude but different sign can be found (-6 and +6, -7 and +7 and so on...) this leaves the sum with only different values being -2-3-4-5 which equals to -14.
On the other hand we could sum the same indice of each given series such as:
(-2+6) + (-3+7) + (-4+8) ... to infinity.
On the second sum, each pair of indice equals to 4 and since there are infinite integers the sum would be 4 * ∞ = ∞.
We could create a generalised rule for the sum of such two finite series but in this example they both go to infinity. My question now is, which one is the true answer to our question? How could we properly represent the summation of such two series?
The series that you showed $$ -6+6-7+7-8+8-\ldots $$ does not converge, as you will be oscillating around $0$ like $$ -6, 0, -7, 0, \ldots $$ However, as you have seen you can get your sum to do a lot of bizarre things by changing the order in which you add terms together. This is because infinite sums are not guaranteed to keep the same value when you shuffle them around. For conditionally convergent sums this would be the Riemann Rearrangement Theorem