This is the problem w infinite sums and calculus...people don't know all the rules, and make folly out of the work done by geniuses before them.
You cannot add infinites like numbers, because they're not numbers! We've found cheeky ways of representing infinity w finite numbers, but infinity , itself, is beyond what our finite minds can truly imagine.
Either way, when you Transpose the infinite concept into our finite world, finite operations work with the finite representation.
Assume infinite sum of numbers (1+2+3+4...) IS equal to -1/12 (we can assume this because we proved it is. Like we proved 2+2=4, or other convergent infinite sums). Let this equal Z
Using the new finite concept, Z+Z is -1/6, which is the same as 2Z..true!
Z+Z =/= Z+Z+Z...Z because you're mixing together the finite and infinite worlds.
2Z=/= nZ except when n=2. This is true using our finite point of view. Not with the infinite point of view. Remember, we only use Z to represent the finite value. If we don't take this value into consideration (we don't use proper infinity aka, we use a REALLY LARGE NUMBER), it breaks apart. Because we aren't using Z anymore.
Also, assuming S=1-1+1-1+1-1+... it's the same deal. Once you STOP, or if you don't use infinity properly, you will get a finite value, which is not infinite, and therefore not S. In the case of Z, if you stop early, you get a huge number, not -1/12. Same here, if you stop randomly, you get S=0 or 1, not 1/2. In infinite space, unfathomable to us, (finite things are unfathomable to us) it would converge (practically, since we're transposing it to our finite world, as with any convergent infinite sum).