Sums of diverging series
August 28th, 2014 at 11:21:06 PM
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Sometime after 1890, an Italian mathematician developed an 'averaging' method for the summation of divergent series.
He said there is a reasonable closed form answer for the S if S=-0+1-2+3-4+5-6+ ......
What is the value for S?
He said there is a reasonable closed form answer for the S if S=-0+1-2+3-4+5-6+ ......
What is the value for S?
August 29th, 2014 at 8:54:58 AM
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Well, Euler claimed it was
1/4
S = 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + ...
= (1-2) + (3-4) + (5-6) + ...
= -(1 + 1 + 1 + 1 + ...)
3S = -(3 + 3 + 3 + 3 + ...)
S = ((1 - 4) + (2 - 5) + (3 - 6)) + ((7 - 10) + (8 - 11) + (9 - 12)) + ...
= -(3 + 3 + 3 + 3 + ...)
= 3S
Therefore, 2S = 0, so S = 0
= (1-2) + (3-4) + (5-6) + ...
= -(1 + 1 + 1 + 1 + ...)
3S = -(3 + 3 + 3 + 3 + ...)
S = ((1 - 4) + (2 - 5) + (3 - 6)) + ((7 - 10) + (8 - 11) + (9 - 12)) + ...
= -(3 + 3 + 3 + 3 + ...)
= 3S
Therefore, 2S = 0, so S = 0
August 29th, 2014 at 9:31:49 AM
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Quote: ThatDonGuyWell, Euler claimed it was
1/4
When adding and subtracting whole numbers, how is the result a fraction???
I invented a few casino games. Info:
http://www.DaveMillerGaming.com/ —————————————————————————————————————
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
August 29th, 2014 at 9:45:37 AM
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https://www.youtube.com/watch?v=w-I6XTVZXww
You have to accept 1-1+1-1+1.... sums to 1/2.
"Then you can admire the real gambler, who has neither eaten, slept, thought nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire for a coup at trente-et-quarante" - Honore de Balzac, 1829
August 29th, 2014 at 11:58:11 AM
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I didn't watch the video yet, but your new spoiler?Quote: thecesspitYou have to accept 1-1+1-1+1.... sums to 1/2.
1/2?
That's the average of any two consecutive sums. Although I agree the answer is either 1 or 0, to average them is wrong.
That's the average of any two consecutive sums. Although I agree the answer is either 1 or 0, to average them is wrong.
That said, here's
S = Negative infinity.
Which allows S = 3S
Which allows S = 3S
I invented a few casino games. Info:
http://www.DaveMillerGaming.com/ —————————————————————————————————————
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
August 29th, 2014 at 12:48:29 PM
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There's a bunch of similar videos from Numberphile on the subject, and it goes into the Riemann Zeta function quite well.
The -1/12 answer has been used in physics quite nicely. Even tough it blows my mind.
The -1/12 answer has been used in physics quite nicely. Even tough it blows my mind.
"Then you can admire the real gambler, who has neither eaten, slept, thought nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire for a coup at trente-et-quarante" - Honore de Balzac, 1829
