Mathematicians can do some amazing things, but some have been accused of living too much in the world of theory rather than fact. Here's a neat video:
Basically, it says that:
1-1+1-1+1-1+1-1+1... = ½
1-S = 1-(1-1+1-1+1-1+1-1+1...)
1-(1-1+1-1+1-1+1-1+1...) = 1-1+1-1+1-1+1... = S
1-S = S
1 = 2S
S = ½
Pretty straightforward, right? Plus we know the series oscillates between 0 and 1, so a number of 1/2 makes sense (as it would be the average, as it were).
Not so fast, Mathicus Maximus!
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The issue here lies in treating S like a natural number; their premise in finding S assumes that S indeed exists and is not simply conceptual. This is a problematic assumption, which I'll demonstrate.
So, let's do us some math wrangling. Buckle up; it's math time!
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First, let's take 2S. In the video, this is treated like 2*S, thus we can solve:
2S = 1
S = ½
OK, all fair and good. Or is it? Let's do some fandangling with 2S and see what we can see.
S = 1-1+1-1+1-1...
2S = 2(1-1+1-1+1-1...)
2S = 2-2+2-2+2-2...
OK, here's the fun part:
2-2+2-2+2-2... = (1+1)-(1+1)+(1+1)-(1+1)...
(1+1)-(1+1)+(1+1)-(1+1)... = 1+1-1-1+1+1-1-1...
1+1-1-1+1+1-1-1... = (1+1-1-1)+(1+1-1-1)+(1+1-1-1)...
(1+1-1-1)+(1+1-1-1)+(1+1-1-1)... = (1-1+1-1)+(1-1+1-1)+(1-1+1-1)...
(1-1+1-1)+(1-1+1-1)+(1-1+1-1)... = 1-1+1-1+1-1+1-1...
And, by definition:
1-1+1-1+1-1+1-1... = S
2S = S
If this is true (which it is, based on the definition of S and the logic used in the video), then we have:
1-S = S = 2S
1-S = 2S
1 = 3S
S = ⅓
We could get any number of possibilities for S as logic dictates that nS = S (n being 1 or an even integer, at least; haven't tried it with odd integers).
The video treats the equation 1-S = S like we can algebraically manipulate it, but we can't do that and still get real results. This is because they assume that there exists a value S and then go about using that value, but they never actually prove that S exists. If we try the same thing with other abstract concepts/numbers, like 0, we get:
4(0)/0 = 8
0/0 = 8/4 = 2
0 = 2(0)
Now to substitute 0=2(0) in our original equation. We still haven't committed any math ills, here.
4(2(0)) = 8(0)
2(0) = 8(0)/4
Now let's substitute again.
2(0) = 8(2(0))/4
0/0 = 16/8 = 2
1 = 2
In this way, we have "proven" that 1 = 2.
Since this is false, one or more of our premises or calculations must be false. We know that it's the division by 0; you can't take 0/0 = 1. The premise from the video commits a similar math sin. Namely, it assumes that S+S = 2S (that S is a number which can be added like that).
However, we have shown that S+S = S, and is in keeping with how the sequence is defined. We cannot simply treat S as a number or we run into the same issue we did above with 0. The defined definition of S will not allow us to simply manipulate it like an algebraic variable. We arrived at the conclusion that 1-S = S through non algebraic means; it was part of the concept of what S is. Similarly, we arrived at the equally valid conclusion that 2S = S. There is obviously more going on here than simple addition or multiplication when we perform arithmetic with it. Therefore, we must either reject the premise that S = ½, or we must accept that S = ⅓, ¼, ⅙, or any other value we can manipulate it to give.