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Old 2020-03-26, 19:27   #3
retina
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"The unspeakable one"
Jun 2006
My evil lair

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Quote:
Originally Posted by Dr Sardonicus View Post
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
Very good.

If we assign:

A = 1 + 1 + 1 + 1 + 1 + 1 + ...
B = 1 + 2 + 3 + 4 + 5 + 6 + ...

Then B - A = B, since pairwise subtraction gives 0 + 1 + 2 + 3 + ... = 1 + 2 + 3 + ...

Therefore A = 1 + 1 + 1 + 1 + 1 + 1 + ... = 0
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