Counter-intuitive Summation

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Re: Counter-intuitive Summation

#41  Postby ughaibu » Jan 23, 2014 1:37 pm

JSS wrote:So, you can't explain it?

It seems to me pretty obvious that if you add something to anything, you necessarily have something different, else you didn't really add anything.

Thus for n!=0, f{x}+n =/= f{x}
Of course I can explain it, but as I would do so as is explained at the link I posted, I needn't bother. Click the link.
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Re: Counter-intuitive Summation

#42  Postby JSS » Jan 23, 2014 1:44 pm

Matt_B wrote:
JSS wrote:How do you measure "size" such that they are the same size?


http://en.wikipedia.org/wiki/Aleph_number

I prefer cardinality, as it's much less ambiguous.

Well so far, that seems to have led to a lot of confusion. I prefer to use a "standard infinity" such as
infA ≡ [1+1+1+...+1]

Then one can use Edwin Hewitt's hyperreal algebra and everything remains logical, no paradoxes arise.
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1+2+3+4+... = -1/12

#43  Postby EvenAdam » Feb 06, 2014 7:26 pm


!
GENERAL MODNOTE
This post was the first post in a duplicate thread devoted to the same topic. Threads now merged. Happy posting in your new, merged thread.


Calilasseia.


From Numberphile...

(The link in the video contains a more satisfactory proof of 1-1+1-1+... = 1/2)

So, is the infinite sum of all natural numbers equal to -1/12, or are there subtleties which make it more ambiguous?
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Re: 1+2+3+4+... = -1/12

#44  Postby Sityl » Feb 06, 2014 7:30 pm

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Re: 1+2+3+4+... = -1/12

#45  Postby surreptitious57 » Feb 06, 2014 7:59 pm


1-1+1-1+1-1 AD INFINITUM IS EITHER 0 OR 1 NOT 0.5 [ THE AVERAGE ]

IT IS NOT POSSIBLE TO GET A FINITE ANSWER FROM AN INFINITE SUM

INFINITY CANNOT BE WRITTEN AS A NUMBER BECAUSE IT IS UNKNOWN
A MIND IS LIKE A PARACHUTE : IT DOES NOT WORK UNLESS IT IS OPEN
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Re: 1+2+3+4+... = -1/12

#46  Postby hackenslash » Feb 06, 2014 8:02 pm

A mathematician and a physicust beg to differ, regardless of font size. As Hawking said, you can't really argue with a mathematical theorem.

Oh, and the reason infinity can't be written as a number is because it isn't a number.
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Re: 1+2+3+4+... = -1/12

#47  Postby EvenAdam » Feb 06, 2014 8:18 pm

surreptitious57 wrote:
1-1+1-1+1-1 AD INFINITUM IS EITHER 0 OR 1 NOT 0.5 [ THE AVERAGE ]

IT IS NOT POSSIBLE TO GET A FINITE ANSWER FROM AN INFINITE SUM

INFINITY CANNOT BE WRITTEN AS A NUMBER BECAUSE IT IS UNKNOWN

S1 = 1 - 1 + 1 - 1 + ...

1 - S1 = 1 - (1 - 1 + 1 - 1 + ...)

1 - S1 = 1 - 1 + 1 - 1 + 1 - ... = S1

1 - S1 = S1

1 = 2S1

1/2 = S1
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Re: 1+2+3+4+... = -1/12

#48  Postby EvenAdam » Feb 06, 2014 8:51 pm

Just saw that this exact thread already exists - Sorry!
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Re: 1+2+3+4+... = -1/12

#49  Postby zoon » Feb 06, 2014 9:07 pm

I always thought something was fundamentally wrong with the universe
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Re: 1+2+3+4+... = -1/12

#50  Postby Sityl » Feb 06, 2014 9:11 pm

EvenAdam wrote:
surreptitious57 wrote:
1-1+1-1+1-1 AD INFINITUM IS EITHER 0 OR 1 NOT 0.5 [ THE AVERAGE ]

IT IS NOT POSSIBLE TO GET A FINITE ANSWER FROM AN INFINITE SUM

INFINITY CANNOT BE WRITTEN AS A NUMBER BECAUSE IT IS UNKNOWN

S1 = 1 - 1 + 1 - 1 + ...

1 - S1 = 1 - (1 - 1 + 1 - 1 + ...)

1 - S1 = 1 - 1 + 1 - 1 + 1 - ... = S1

1 - S1 = S1

1 = 2S1

1/2 = S1


Can you explain this step to me? I don't see how 1 - S1 = S1. When you subtract 1 from each side, shouldn't you know have -1 + 1 - 1 + 1 - 1 + ... ? I don't think you're able to just take a one off of "the end" since there is in fact no end.

(could be very wrong on this, open to being shown how)
Stephen Colbert wrote:Now, like all great theologies, Bill [O'Reilly]'s can be boiled down to one sentence - 'There must be a god, because I don't know how things work.'


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Re: 1+2+3+4+... = -1/12

#51  Postby Pulsar » Feb 06, 2014 9:17 pm

EvenAdam wrote:Just saw that this exact thread already exists - Sorry!

I've asked for a merge :cheers:
"The longer I live the more I see that I am never wrong about anything, and that all the pains that I have so humbly taken to verify my notions have only wasted my time." - George Bernard Shaw
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Re: 1+2+3+4+... = -1/12

#52  Postby EvenAdam » Feb 06, 2014 9:17 pm

Sityl wrote:
EvenAdam wrote:
surreptitious57 wrote:...

S1 = 1 - 1 + 1 - 1 + ...

1 - S1 = 1 - (1 - 1 + 1 - 1 + ...)

1 - S1 = 1 - 1 + 1 - 1 + 1 - ... = S1

1 - S1 = S1

1 = 2S1

1/2 = S1


Can you explain this step to me? I don't see how 1 - S1 = S1. When you subtract 1 from each side, shouldn't you know have -1 + 1 - 1 + 1 - 1 + ... ? I don't think you're able to just take a one off of "the end" since there is in fact no end.

(could be very wrong on this, open to being shown how)

The idea is not to subtract 1 from the series, but to subtract the series from 1. So, the new series would begin with 1 and the following terms would be the old series but with all the signs inverted(distributive property), resulting in the original series.

However, I just noticed this thread exists: http://www.rationalskepticism.org/mathe ... 43441.html

There is some discussion about the problems with using that kind of intuitive logic with infinite series.
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Re: 1+2+3+4+... = -1/12

#53  Postby Sityl » Feb 06, 2014 9:54 pm

Okay, so if my understanding is correct (and it probably isn't), you can't add the series in a traditional manner, because it doesn't converge, it diverges. So, in order to find the convergence, you actually look backwards on the x axis, until it intersects the y axis at -1/12.
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Re: Counter-intuitive Summation

#54  Postby Pulsar » Feb 06, 2014 11:54 pm

I found two more advanced discussions where this sum comes up.

The first is a video from Dr James Grime (one of the mathematicians in the Numberphile videos), where he discusses the The Riemann Hypothesis. The relevant bit is the analytic continuation of the Riemann Zeta function, from 8:38 to 11:40.



The second link is a blog post from Terence Tao:

The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation
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Re: Counter-intuitive Summation

#55  Postby surreptitious57 » Feb 07, 2014 4:47 am

Ramanujan wrote:
Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you
similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series
and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 +
2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I
dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which
I proceed in a single letter

So this equation is over a hundred years old and was solved by a mathematician with no formal training who was arguably
the greatest ever. Had that been mentioned in the video then I would have accepted it without question. Argument from authority is of course not infallible but Ramanujan was a complete one off. He even managed to surpass Hardy who was regarded as one of the greatest mathematicians of his generation. He is to mathematics what Einstein is to physics. So he
ought to be a household name but famous mathematicians tend to be completely unknown outside of mathematics which
is rather unfortunate because famous scientists by comparison are not
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Re: Counter-intuitive Summation

#56  Postby jaydot » Apr 11, 2014 5:47 pm

i read somewhere that the "graham" number is the largest number in the universe. "bollox", i said as i added a "1".
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Re: Counter-intuitive Summation

#57  Postby newolder » Jan 30, 2018 4:46 pm

... and another continuation (ahem!) of this topic posted by Mathologer:
I am, somehow, less interested in the weight and convolutions of Einstein’s brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops. - Stephen J. Gould
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