> Obviously this is pure nonsense.

This is begging the question. Why can't 1 + 1 + ... = 0?

Also, I wouldn't be so sure that S2 == S1. You can't re-arrange infinitely many terms in an infinite series and still be guaranteed the sum is the same.

S2 is identical to (0 + S1). Are you suggesting that (0 + S1) != S1 ?

Precisely. Adding 0 to the front of an infinite series is shifting every term by one to the right. It's not clear that shifting terms in series keeps the sum the same. For instance, re-arranging infinitely many terms in conditionally convergent infinite series changes the sum.

> Adding 0 to the front of an infinite series ... not clear that ... keeps the sum the same

I don't know if I would go that far... but I agree with the general spirit of your comment. Which is also the point of my original post. If you think that my appending a zero calls my proof into question, the proof presented in the video takes far more dubious and horrific liberties.

I don't know for certain if (0 + S1) ?= S1, but infinite series require care. Consider this:

Let S1 = 0 + 0 + 0 + ... = 0 Then surely, S1 = (1 - 1) + (1 - 1) ... = 1 - 1 + 1 - 1 + ... -1 + S1 = -1 + 1 - 1 + 1 - 1 ... = (-1 + 1) + (-1 + 1) ... = 0

But then -1 + 0 = 0

Your error is here, equality is wrong: -1 + 1 - 1 + 1 - 1 ... = (-1 + 1) + (-1 + 1) ...

In first series, there are 2 different elements (1 and -1) and series can end at any one of them rendering the end result of the sum uncertain. On second one there is only one element - (-1 + 1) which is 0, so wherever you end it the result is always the same.

My whole point was there are operations that work in finite mathematics that don't work on infinite series, so yes, I didn't prove mathematics inconsistent, I just proved grouping is illegal for divergent infinite series :)

To be clear, yes

(1 - 1) + (1 - 1) + ... != 0 + 0 + ... either

> To be clear, yes

> (1 - 1) + (1 - 1) + ... != 0 + 0 + ... either

I don't see why. I understand the sets themselves are not equal, but the sum of the elements of those sets is at any given index.

It's not obvious that S2 == S1.

I can see some differences: for any finite N > 0, it's false that the sum of the first N terms of S2 is the same as the sum of the first N terms of S1. And what do we know about the infinite sum? Maybe you're right, but you'd have to prove it; they are definitely not equal "by definition"!

You're right - "infinity" is more of a type than an actual value - S1 and S2 have different counters.

This is actually a much better calculation than the one in the video, because it doesn't rely on undefined == 1/2. But it does show that some of the other steps in the video are bogus too, so it's even worse than I thought. I thought that just the 1-1+1-1+1... = 1/2 was the problem (the equivalent of the division by zero in all those 1==2 proofs).

That 1/2 instantly bothered me. Who decides that average of fluctuation between 2 values is their sum? I understand how it works out to call the sum 1/2, but only if the definition of "sum" is changed. More accurate description of number would be something like "convergence of average of sums of Nth element, where N is going to infinity" (call it X). And when looked at it like that, we see that other numbers, like 1/4 are understandable too. But the -1/12 is not even X. It is a number that we get by manipulating X of one series with X of a couple of other series by applying some arbitrary rules that doesn't necessarily make the number mean what the mathematician in the video thinks it means. I am sure the number can be made useful in certain problem, but to call it "sum of infinite series" is just plain wrong.