I looked it up, it's called Grandi's series. There are at least three proofs offered on this page:
https://en.wikipedia.org/wiki/Grandi%27s_series
Here's one easy to understand proof:
Proof through 1 / x series
The series:
1/x = 1 - (x-1) + (x-1)^2 - (x-1)^3 + (x-1)^4 - ...
Is fairly easy to prove. First, multiply everything by x. On the left side, this makes 1, and on the right side we'll represent x as (x - 1) + 1. Multiply the series by (x - 1) and 1 separately and add the two together.
1 = (x-1) - (x-1)^2 + (x-1)^3 - (x-1)^4 + ... + 1 - (x-1) + (x-1)^2 - (x-1)^3 + ...
All terms except 1 cancel out, leaving:
1 = 1
Applying this series to 2 gives:
1/2 = 1 - 1 + 1 - 1 + 1 - ...
That's the critical part, if that part works, the rest of the -1/12 proof is just basic arithmetic carried out on the different series.
It's not legal to "cancel out" things like that in an infinite series. (1 - 1 + 1 - 1 + 1 - ...) is both equal to zero and one, using that "cancel out" technique:
a) If you "cancel out" the first number with the second, the third with the fourth and so on, the sum becomes (1 - 1) + (1 - 1) + ... = 0.
b) If you cancel out the second number with the third, the fourth with the fifth and so on, the sum becomes 1 - (1 - 1) - (1 - 1) - ... = 1.
Since a series cannot be equal two different numbers, it becomes obvious that this "canceling out" technique breaks mathematics, a la division by zero (which can also be used to "prove" various fun equations like 1 = 2).
EDIT: It seems that when they mean the
https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation when they say that it equals 1/2, which is kind of a different thing.
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Topic: Things that made you go "WTF?" today o_O (Read 14951408 times)