October 30th, 2015, 08:14 PM  #11 
Global Moderator Joined: Dec 2006 Posts: 21,036 Thanks: 2273  You seem to be misreading the graph. The parabola shown is for y = x²/2  1/12.

October 30th, 2015, 08:24 PM  #12 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,691 Thanks: 2670 Math Focus: Mainly analysis and algebra 
I've just watched your video, which while interesting does contain one glaring problem. You have arbitrarily decided that your partial sums should be symmetrical and, even worse, that these finite "negative" partial sums should influence the positive infinite sum. Admittedly, the methods of assigning values to these sums also make assumptions, but they are minimal assumptions designed to mirror the approach we take with convergent sums. 
October 31st, 2015, 04:15 PM  #13  
Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2  Quote:
I had a good reason to choose symmetry. If you look at the graph for x(x+1)/2 but just consider positive values of x, and then you 'remove the diverging part', then nothing remains. This matches what you would expect from the partial sum expression. The fixed part of the partial sum expression is what is called the 'limit' for a 'converging' series. It is my understanding that these methods are trying to get this value for a diverging series, and I guess it might not always be so easy to find the fixed part of the partial sum expression. The reason I wanted symmetry was to ensure that methods that 'remove the diverging part' no longer get this value of 1/12, which they would not get if they were just considering the positive values of x. Symmetry allows these other methods to arrive at the the result of zero, instead of getting this 1/12 value resulting from the skew. Hence they no longer contradict the value obtained from the partial sum expression. This is the same reason I wanted symmetry of the Zeta function, like this would produce: $$\sum_{n=1,2,3...}\frac{s}{s.n^s}$$ Last edited by Karma Peny; October 31st, 2015 at 05:13 PM.  
October 31st, 2015, 06:18 PM  #14  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,691 Thanks: 2670 Math Focus: Mainly analysis and algebra  Quote:
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These techniques obey rules developed from the rules of arithmetic and analysis by discarding the rules of the finite realm that demonstrably no longer apply in the infinite realm. If you wish to attack the results, you'll have to find flaws in the formulation of the rules, not in the results that spring from them.  
October 31st, 2015, 09:30 PM  #15  
Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2  Quote:
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No I don't. I claim that if a series is endless it should treated as being endless, not as if it were a fixed value. Finite series have fixed values, endless series don't. They should be treated differently.  
October 31st, 2015, 10:49 PM  #16  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,691 Thanks: 2670 Math Focus: Mainly analysis and algebra  Quote:
You are also fixed on the concept of evaluating the sum. This is expressly what the ideas I am talking about do not do. They assign a value to a series, it is not necessary to believe that it has any link with evaluating the series. Quote:
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In the statement of the rule that I gave, there are two (related) series $a_0+a_1+a_2+\ldots$ and $a_1+a_2+\ldots$. The "rest of the series" phrase was intended to highlight the relation between the two, no more. The basic assumption here is less mathematical than lexical. Given a mathematical object of a certain structure, we assert that we can give it a value in the extended real numbers. This value need not have any relation to evaluating the sum that the object looks like, we only insist that certain rules are followed in generating the value that we assign to the object. Quote:
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This whole subject can be seen as an extreme finitist approach to infinite sums  we explicitly abandon any attempt at evaluating an infinite sum inf favour of finding a finite value by which the sum can be represented in finite arithmetic. This is in direct contrast to our usual way of viewing convergent sums where we are in the habit of claiming that the limit is the evaluation of the infinite series. Last edited by v8archie; October 31st, 2015 at 10:53 PM.  
November 1st, 2015, 04:49 AM  #17 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,691 Thanks: 2670 Math Focus: Mainly analysis and algebra 
I should also add that a "reasonable" value for a convergent series is it's limit. This idea was a reason for the vagueness of that sentence. It occurs to me that we might compare this theory with calculus under the hyperreal numbers. Both are formalisations of nonrigours methods used in the past. I'd claim that this one is superior to the hyperrals, because of the way definitions are stated. The calculus incorporates an error correction into the definitions. 
November 1st, 2015, 11:00 AM  #18  
Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2  Quote:
Why not avoid these problems and simply treat an endless object as an endless object (like I do), rather than assume 'infinity' must exist? I can't believe you are saying this. I keep stressing over and over that an endless series (where the endless terms are nonzero) cannot have a sum. If anything, I am fixed on NOT evaluating the sum Quote:
Why 1/12 is a gold nugget: In this video he says about the sum of the powers of natural numbers: "Why did I skip squares... the answer is more surprising... for all even values you actually get zeroes, and for odd values you get some rational number" But to me this is not surprising at all. If you consider the graph of the partial sum expression, then for even values you will get a reflection about the yaxis but for odd powers you will not. He also says: "We are not content with just saying that there is some magic over there, there is magic, but we always want to explain it" And: "does it mean that in some sense it is, there is a context in which this sum, this infinite sum, is mysteriously minus one twelfth? I'm not sure." It appears to me that mainstream mathematics regard this 1/12 value as being shrouded in the mystery and magic that comes from using methods that claim to work with 'infinity' in some way. Because if you then proceeed to use it, you will not understand what you are doing. And if you don't proceed to use it, then it is of no value. Quote:
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Being a habit does not make it correct. Last edited by Karma Peny; November 1st, 2015 at 11:05 AM.  
November 1st, 2015, 05:52 PM  #19  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,691 Thanks: 2670 Math Focus: Mainly analysis and algebra  Quote:
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I don't agree with this. I think that it's Numberphile's way of simplifying the idea for a more general audience. Either that, or the physicists interviewed don't understand the mathematical basis of the ideas. The idea of "removing the infinite parts" suggests that you have been adding the elements of the sum. My main reference on this is Carl Bender's lectures, where he does say that adding up the terms of infinite series is stupid. Quote:
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We are going to assign a value on the Riemann Sphere $s$ to $1+1+1+\ldots$ using generic summation. Thus $$\begin{aligned}s &= S(1+1+1+...) \\ s &= 1+ S(1+1+1+...) &\text{by the first rule I gave above} \\ s &= 1 + s \\ \implies s = \infty\end{aligned}$$Now, there's a lot of "fudge" in that. But the point is not in how we get to that answer, but the fact that we accept the answer that we get. Essentially it means that "this technique is not powerful enough to assign a finite value to this series". It turns out that analytic continuation is powerful enough to do that, and gives us the value ${1 \over 2}$. In essence, I believe that you need to stop concentrating on finite partial sums. They work well as a tool for convergent series, but are not powerful enough to assign a finite value to divergent series. I also believe that you need to accept mathematics for what it is, rather than for how it can be interpreted. Last edited by skipjack; November 2nd, 2015 at 12:16 PM.  
November 2nd, 2015, 11:27 AM  #20  
Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2  Quote:
Nice attempt at understanding the distinction. You completely ignored the 'paradoxes' I mentioned and went off to talk about sets. Quote:
After watching that Numberphile video (link in my previous post), I decided to derrive the partial sums for $$1^2 + 2^2 + 3^2 + 4^2 +...$$ and for $$1^3 + 2^3 + 3^3 + 4^3 + ...$$ then I plotted them on a graph. I also took the definite integrals of these expressions between 0 and 1, which is where the relevant part of the 'skew' occurs. Guess what, I got zero and plus 1/120 just like in the video. So, either it is one huge coincidence that these figures match or the Extreme Finitism approach has correctly identified where these ‘mysterious’ values are coming from (& it surely isn’t infinity!). You can keep your mysteries and your belief in 'infinity'  you must do whatever makes you happy, but I prefer to understand what I'm doing. It has been enjoyable discussing this with you, I hope I don't sound too disgruntled. We all must live with our own beliefs and not chastise others for their beliefs. Many thanks once again, and goodbye. Last edited by Karma Peny; November 2nd, 2015 at 12:24 PM.  

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1/12, 1 or 12, infinity, ramanujan, relationship, series, zeta 
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