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2020-03-26, 19:06   #2
Dr Sardonicus

Feb 2017
Nowhere

22·3·317 Posts

Quote:
 Originally Posted by retina Is (1 + 1 + 1 + 1 + 1 + 1 + ...) less than (1 + 2 + 3 + 4 + 5 + 6 + ...) ? It would seem not. Because 1 + 2 + 3 + 4 + 5 + 6 + ... = -1/12 https://www.youtube.com/watch?v=w-I6XTVZXww I'm not sure if making a comparison like the above is valid. Perhaps I misunderstand comparisons of infinite sequences?
Of course, the series as written do not converge, so simply taken at face value the question is nonsense.

One can assign values to the sums by misusing formulas. The value -1/12 assigned to 1 + 2 + 3 + ... is a case in point. We have

$\zeta(s)\;=\;\sum_{n=1}^{\infty}n^{-s}\text{, when }\Re(s)\;>\;1$

The zeta function is defined at s = 0 and at s = -1 (though is not given by the above series at those points), taking the values -1/2 and -1/12, respectively.

Cheerfully disregarding the invalidity of the formula, mindlessly plugging in s = 0 gives

1 + 1 + 1 + ... ad infinitum = -1/2

and plugging s = -1 into the formula gives

1 + 2 + 3 + 4 + 5 + 6 + ... ad infinitum = -1/12.

And -1/2 < -1/12.

:-D

Last fiddled with by Dr Sardonicus on 2020-03-26 at 19:09 Reason: Rephrasing