1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

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Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

Post by Dutti on Tue Jan 14, 2014 3:23 pm

@Ion Creanga wrote:
@Dutti wrote:
@Ion Creanga wrote:i'm student at mathematics in Romania, if my math analysis teacher would see that he would get crazy Laughing
the proof is wrong because it uses false assumptions... That infinite sum (1-1+1-1...) makes 1/2 in his opinion.. in Fact that sum has no result... it is a series with the general with the general number Xn=(-1)^(n+1), which is not a convergent series... because you can take two substrings of the partial sum Sn=  that have different limits (1 or 0), so the series not being convergent it can't have the sum a float number. Read some theory about series and you'll understand easy...  If this result would be true all the convergence theories and math analysis would be fcked ...
you can obtain sh*ts like this if you make false assumptions (like dividing an expression with 0)

Sorry for my english...

Using the definition of limit is no different from finding the sum of the series by adding term by term. So it is obvious that the series is divergent in that case.

But I read somewhere that there is a type of Mathematics that allows you to define the sum treating it as "what if the sum does exist then what should it be?".

Although I don't remember the name of that type of Maths, but that is what exactly being applied in the proof that leads to the sum being 1/2.

After all, since you are a student of Maths, don't you think it is troublesome to define the alternating series equal to some S? To me that "S = " is troublesome; so applying the Maths that I just mentioned treats the proof as a "what if" case. It's like treating i = sqrt(-1).

There is a little bit of insight to that type of Maths explained by Dr James Grimes (a Ph.D, btw) in one of Numberphile's videos. In it, he explains that there is a type of sum that is almost like a limit but nevertheless NOT the same as limit. He calls it "pseudo-limit" and that is what, as he explains, used to proof that the sum is 1/2.

Also, to justify that the answer is 1/2 in the REAL world (instead of pure maths), consider what would happen to the proof in the OP if the sum = 1/2 were completely false. Quantum Mechanics would fall apart because the proof that the infinite series (in the OP) = -1/12 depends on the assumption that the sum of (-1)^(n+1) = 1/2
I used the definition of limit just to proof that the partial sum string is not a convergent string, therefore the series can't have a result (not sure if you studied series in university to know what partial sum is) ... You can prove that this alternating series is not convergent in many ways, i gave one...
Regarding the S, like you said it's wrong to assume that S (real number) is the result of that infinite sum and make operations with S, because S does not exist ...





lim of S_n as n approaches infinity...that's pretty basic; treating the partial sums as an infinite sequence that is.

You don't need a Math major to understand that since it's learned in first year anyway. Even engineering students understand that.

However, most people do not realize that the limit definition of partial sums taking n to infinity is the same as adding the terms in the series itself one by one, and I'm not sure you've realized this.

Regarding the bold sentence I did not claim that. I guess you misunderstood my saying "what if it DOES exist". Regardless of what you understand, you seem to have overlooked Dr Grimes's explanation. That's pretty bold.

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Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

Post by Ion Creanga on Tue Jan 14, 2014 9:34 pm

"lim of S_n as n approaches infinity...that's pretty basic; treating the partial sums as an infinite sequence that is."

yes, of course...

"However, most people do not realize that the limit definition of partial sums taking n to infinity is the same as adding the terms in the series itself one by one, and I'm not sure you've realized this."

why do you assume that? once you learn series it's a pretty easy concept..
of course i realized that taking the partial sum tu infinity it's like adding the terms of the series, my point was just to give a simple explanation for the divergence of that series... Too many talk over nothing...

"Regarding the bold sentence I did not claim that. I guess you misunderstood my saying "what if it DOES exist". Regardless of what you understand, you seem to have overlooked Dr Grimes's explanation. That's pretty bold."

i didn't look carefully over that explanations because i prefer pure mathematics with the real number axiomatic system...



Let me give you a beautiful math logic problem:

-Let R be the set of all sets that are not members of themselves... Does R contain R??  Very Happy

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Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

Post by Dutti on Wed Jan 15, 2014 5:22 am

@Ion Creanga wrote:"lim of S_n as n approaches infinity...that's pretty basic; treating the partial sums as an infinite sequence that is."

yes, of course...

"However, most people do not realize that the limit definition of partial sums taking n to infinity is the same as adding the terms in the series itself one by one, and I'm not sure you've realized this."

why do you assume that? once you learn series it's a pretty easy concept..
of course i realized that taking the partial sum tu infinity it's like adding the terms of the series, my point was just to give a simple explanation for the divergence of that series... Too many talk over nothing...

"Regarding the bold sentence I did not claim that. I guess you misunderstood my saying "what if it DOES exist". Regardless of what you understand, you seem to have overlooked Dr Grimes's explanation. That's pretty bold."

i didn't look carefully over that explanations because i prefer pure mathematics with the real number axiomatic system...



Let me give you a beautiful math logic problem:

-Let R be the set of all sets that are not members of themselves... Does R contain R??  Very Happy

Indeed, it's nothing important...lol
I only mentioned it cause you thought I didn't know that simple definition.

Dr Grime is not a physicist. He's a mathematician.
http://www.youtube.com/watch?v=PCu_BNNI5x4

That paradox doesn't have any significance in today's Math, at least not that I am aware of.

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Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

Post by Ion Creanga on Wed Jan 15, 2014 8:50 pm

Yea, the paradox doesn't have any significance today, but back than, during the beginnings of the 20 th century, it was enough to blow away Cantor's "naive" set theory (that you can build a set , a collection of objects with a property)...

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Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

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