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> 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
by zeroer 10y ago
> 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.
- whack 10y agoS2 is identical to (0 + S1). Are you suggesting that (0 + S1) != S1 ?
- ebola1717 10y agoI 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
- cLeEOGPw 10y agoYour 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.
- ebola1717 10y agoMy 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
- cLeEOGPw 10y ago> 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.
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- zeroer 10y agoPrecisely. 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.
- whack 10y ago> 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.