Originally posted by Boxing Pythagoras
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1 + 2 + 3 +... = -1/12
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Originally posted by Leonhard View PostTo be fair to the point Pixie is trying to make, I do make a hidden assumption that derivation that S is a definite number. So the logic of the derivation should start with the statement 'If S has a definite value, we can find it from Abel's summation of another series, by the following number of steps'.
Still, the result is entirely consistent with what you get from Ramanujan Summation and analytic continuation.
1 + 2 + 3 + 4 .... + n ... + infinity = infinity
Proof 1
Any finite number added to infinity is infinity, so when summing 1 + 2 + 3 + 4 .... + n ... + infinity we can add all the finite terms to the final term, and we get infinity.
Proof 2
There is a formula for the sum S = 1 + 2 + 3 + 4 .... + n which is readily derived.
For the series, we can pair up values, starting at the ends. Pair the first with the last (1 + n), the second with the last but one (2 + n-1), etc. Each pair therefore has a total of n + 1. Further, there are n / 2 pairs. The total of all the pairs is therefore:
S = (n+1).n/2
You can try this for smaller values of n to confirm, or just look at these websites.
http://www.wikihow.com/Sum-the-Integers-from-1-to-N
http://mathforum.org/library/drmath/view/57919.html
This is a well known formula, and I remember being shown this proof at school. The fact that the calculation in the OP does not use it shows there is something suspect going on there.
So as n tends to infinity, what does S tend to?
Well, infinity + 1 is infinity. Infinity over 2 is infinity. And infinity times infinity is infinity. See here if in doubt:
http://scienceblogs.com/goodmath/200...-not-a-number/
http://www.ditutor.com/limits/infinity.html
http://www.mathsisfun.com/numbers/infinity.html
Even if you dispute that infinity works like that, it is readily apparent that S is greater than n for any n greater than 1.
So if n = infinity, S = infinity.
Now I invite you to find an error in my reasoning. If you can find none, then you have to acknowledge that either S is both infinity and -1/12 or that it is not -/12.My Blog: http://oncreationism.blogspot.co.uk/
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Originally posted by Boxing Pythagoras View PostYour analogy is a bad one, since your claim of 2+2=5 would not be consistent with everything else in mathematics, whereas Ramanujan summation is consistent. Again, mathematics does not conform itself to the intuitions of the general population. Ramanujan summation is every bit "maths as it is usually understood" as is any other infinite summation.
Again, the very definition given in the Wolfram article tells you all that you need to know. Ramanujan summation is defined by a particular application of the summation operator. It is every bit as much a summation as any other application of the summation operator, despite the counterintuitive result.
I had said...
Regardless, it is still trivially true that the Distributive Property is applicable regardless of the number of terms being summed.
...to which you had replied...
Then you will have no problem walking us through it in this example.
As such, I was under the impression that you did not believe that the Distributive Property was applicable to this summation, and so I addressed that.
Again, you are mistaken, here. Infinity is not a number, and you were therefore wrong when you claimed, "S is infinity." Multiplying infinity by 4 is not defined. Subtracting infinity from infinity is not defined. You cannot perform numerical operations on things which are not numbers. However, the number from Leonhard's equations, S, is an actual number. It is not infinity. It is a well-defined number. We can multiply a well-defined number by 4, and we can subtract from a well-defined number.
Furthermore, multiplying infinity by a finite, non-zero number is defined (but subtracting infinity from infinity is not, which is part of the problem)).
http://www.vitutor.com/calculus/limi..._infinity.htmlMy Blog: http://oncreationism.blogspot.co.uk/
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Originally posted by The Pixie View PostBut it is not consistent with conventional mathematics.
1 + 2 + 3 + 4 .... + n ... + infinity = infinity
Proof 1
Any finite number added to infinity is infinity, so when summing 1 + 2 + 3 + 4 .... + n ... + infinity we can add all the finite terms to the final term, and we get infinity.
Proof 2
There is a formula for the sum S = 1 + 2 + 3 + 4 .... + n which is readily derived.
For the series, we can pair up values, starting at the ends. Pair the first with the last (1 + n), the second with the last but one (2 + n-1), etc. Each pair therefore has a total of n + 1. Further, there are n / 2 pairs. The total of all the pairs is therefore:
S = (n+1).n/2
Now I invite you to find an error in my reasoning. If you can find none, then you have to acknowledge that either S is both infinity and -1/12 or that it is not -/12.
However for those methods capable of summing up divergent series, the result is -1/12.
There exists a broad category of methods for taking divergent series, like the Ramanujan Summation and the Abel Summation, that consistently give the same answers for divergent series (where they have the power to sum these series). That's not to say that there aren't limits. Its required for each of these functions of n that are summed over that they behave analytically at certain points (especially if its to be summed by Ramanujan Summation).
And whether you like the result or not is irrelevant, its an example of a branch of mathematics that ends up explaining how to get finite results from divergent integrals, which pop up all over the place Quantum Field Theory, where we get a whole system of regulators and renormalization to squeeze out finite results (that experiments confirm are correct) from integrals that diverge towards infinity.
What is true is that in Quantum Field Theory its a mystery why this should work. Why are we justified in making a switch to a different way of doing integrals and sums than the way we've been taught in introductory courses? So far no realistic interpretation of this has emerged, and some feel it indicates that Quantum Field Theory lacks something and has yet to be brought to a completely satisfying mathematical basis yet.Last edited by Leonhard; 02-04-2015, 03:01 AM.
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Originally posted by Leonhard View PostI'm really not sure what you're doing here, but you can't use infinity as a number, and you can't express the summation of an infinite series as having been completed that way.
I am not saying you are wrong (in fact I put up a second proof because I thought this was dodgy), but I have found several web sites that support my claim that the sum is infinite, not -1/12, so at this point I am not accepting I am wrong merely because you declare it is.
I think you're arguing a point I've already said. The partial sum of all members up the nth term of that series tends towards infinity, as n goes to infinity.
The partial sum is divergent. However the question wasn't whether this series was divergent, it obviously is, in fact it so obviously divergent that the fact that you can derive a consistent finite result less than 0, which is still meaningful, is so utterly counter intuitive that its hard for many people to swallow.
Where is the error in my calculation?
I maintain that my calculation is superior to yours because it does not assume its conclusion (yours appears to assume S is finite, according to BP), and it does not involve infinity minus infinity, which is undefined.
You said to another poster:
"I know its counter-intuitive, but if you want to convince me that I made a mistake you'll need to be explicit."
Well, if you want to convince me that I made a mistake you'll need to be explicit.
By the way, I asked BP two questions, I would appreciate your own answers:
What do you think 4 times infinity is?
What is infinity minus inifinity?My Blog: http://oncreationism.blogspot.co.uk/
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Originally posted by The Pixie View PostWhy?
However if you want it to constitute a proof - which you seem to by giving it a bolded title 'proof 1' - then it'll have to be stringent and yours simple isn't. I know it probably reflects your intuitive thinking on it, but there it isn't a display of mathematical logic.
I really don't have to say anything more on it. It's simple not a proof.
It may be meaningful in some context, but is it equal to the sum?
I maintain that my calculation is superior to yours because it does not assume its conclusion (yours appears to assume S is finite, according to BP), and it does not involve infinity minus infinity, which is undefined.
At no point do I have to concede to you that I'm subtracting infinity from infinity.
What do you think 4 times infinity is?
What is infinity minus inifinity?Last edited by Leonhard; 02-04-2015, 06:01 AM.
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Originally posted by Leonhard View PostI used the method I used because its easier to follow and explain, and its not actually wrong. The stronger derivation is from Ramanujan Summation, however its much harder to explain how Ramanujan Summation works, but it too produces the exact same value for S, albeit in a different way. So my assumption is not unjustified.
If S is infinite, as my calculation indicates, then your calculation is flawed.
At no point do I have to concede to you that I'm subtracting infinity from infinity.
Did you read the web pages I linked to? Strangely, they agree with me.
Since the infinity you're using isn't a number, neither operation is properly defined.
http://www.ditutor.com/limits/infinity.html
http://www.mathsisfun.com/numbers/infinity.html
http://www.vitutor.com/calculus/limi..._infinity.html
http://tutorial.math.lamar.edu/Class...fInfinity.aspx
Perhaps you can find some links to support your contention that a finite positive number multiplied by infinity is not defined.My Blog: http://oncreationism.blogspot.co.uk/
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Originally posted by Leonhard View PostIn another thread the prospect of infinite sums came up and I wanted to introduce one of the oddest results I've ever seen in math. Its a situation where clearly the partial sums tend towards infinite, and where its possible to develops means of summing up the infinite numbers in such a way that it returns a specific result.
Let's dive right into it and give the result.
This ridiculously counterintuitive result stems from a field called Ramanujan summanation, which is one in a group of summation methods for infinite series that are divergent. Strangely enough some of these methods can be used to sum up the infinite series of integers, and they always end up given the result above.
I'll show you how this result can be derived from something called Abel summation.
Now if we take another series, namely
Then naturally we'd try to solve this by finding its partial sum (summing up all but the leading term) and seeing what this converges to
Obviously the partial sum is divergent as we add more and more terms. So we'll need a different way of summing. The mathematician Abel proposed (though Euler found it first), the following means of handling this type of summation.
This can't sum the first series mentioned, but it is able to sum the second one, giving us...
From this we can determine what series mentioned in the beginning will be, if it has any value at all. This is done simple be substracting the partial sum of one series from the other.
Of course this result can also be found directly, but that requires a stronger method of summation than Abel summation, such as Ramanujan summation. However unsurprisingly it yields the exact same answer. And its not merely a theoretical answer, as sums over all the natural integers occur often in quantum field theory, and in the derivation of the Casimir effect one has to use such a sum.
Therefore this result, along with various strong derivations are included in Advanced Quantum Mechanics course work.
What do you guys think?
My immediate thought was, "just another example of how New Age
thinking has corrupted true science". Enter Evolutionism, stage left.
It's all hogwash, Pixie ... you know, the kind of stuff that Evolutionists love to eat up.
Not much more to say than that.
Jorge
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Originally posted by The Pixie View PostBut you are trying to prove S is finite, and to do that, you are obliged to assume it is finite!
If S is infinite, as my calculation indicates, then your calculation is flawed.
Did you read the web pages I linked to? Strangely, they agree with me.
Some of them are little more than collections of operations and the purported results, one is an amateur homepage explaining various things. None of them actually make the argument that infinity is a number, or that it can properly be treated as one.
I know you're doing a lot of work to claim that the partial sum does diverge. And I don't disagree, it clearly, abundantly obviously diverges! The point of the post wasn't to disprove that it diverges. If that was the point I would be an idiot, a crank, or a crank idiot. The points was that many sums that diverge, can in fact by summed to a finite (and sometimes counterintuitive and surprising) value, and whether we like this fingerspitzengefühl or not, this lies close to the kind math regularly used and wide embraced in the renormalization theory in Quantum Field Theory.
Make of it what you want, but I haven't committed any errors.
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Originally posted by The Pixie View PostI am not saying you are wrong (in fact I put up a second proof because I thought this was dodgy), but I have found several web sites that support my claim that the sum is infinite, not -1/12, so at this point I am not accepting I am wrong merely because you declare it is.
Originally posted by The Pixie View PostBut you are trying to prove S is finite, and to do that, you are obliged to assume it is finite!
If S is infinite, as my calculation indicates, then your calculation is flawed.
The symbol ∞ does not refer to a number. However, there are numbers which are infinite, and operations which are not defined for ∞ are defined for these latter.Last edited by Boxing Pythagoras; 02-04-2015, 02:07 PM."[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
--Thomas Bradwardine, De Continuo (c. 1325)
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Some of them are little more than collections of operations and the purported results, one is an amateur homepage explaining various things. None of them actually make the argument that infinity is a number, or that it can properly be treated as one.
In summary then, I have some web pages you consider to be poor to support my position, you have nothing but your own supposed auithority to support your own assertion.
I know you're doing a lot of work to claim that the partial sum does diverge. And I don't disagree, it clearly, abundantly obviously diverges! The point of the post wasn't to disprove that it diverges. If that was the point I would be an idiot, a crank, or a crank idiot. The points was that many sums that diverge, can in fact by summed to a finite (and sometimes counterintuitive and surprising) value, and whether we like this fingerspitzengefühl or not, this lies close to the kind math regularly used and wide embraced in the renormalization theory in Quantum Field Theory.
Make of it what you want, but I haven't committed any errors.
Originally posted by Leonhard View PostWhat do you think 4 times infinity is?
What is infinity minus inifinity?
Therefore, you have to assume S is not infinite for your argument to make sense.
And yet, intuitively S is indeed infinite, so your assumption is bad.My Blog: http://oncreationism.blogspot.co.uk/
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Originally posted by Boxing Pythagoras View PostYou've given several websites which list operations which can be performed using the infinity symbol, ∞, ...
Originally posted by Boxing Pythagoras View PostYou've given several websites which list operations which can be performed using the infinity symbol, ∞, but I don't believe you've provided any which support your claim that in explicit contradiction to -1/12. Did I miss one?
http://physicsbuzz.physicscentral.co...utely-not.html
http://skullsinthestars.com/2014/01/...ome-would-say/
To put it another way: in a restricted, specialized mathematical sense, one can assign the value -1/12 to the increasing positive sum. But in the usual sense of addition that most human beings would intuitively use, the result is nonsensical.
http://www.slate.com/blogs/bad_astro...ng_result.html
But there is a method called analytic continuation that does. It redefines things a bit, uses different rules that allow for dealing with such things. The mathematicians Euler and Riemann used it to get around the problems of infinite diverging series, and it allowed them to assign the value -1/12 to it. Those rules are self-consistent, logical, and highly useful. In fact, as I pointed out in the previous post, they’re used to great success in many fields of physics. It gets complicated quickly, but you can read more about this here and especially here (that second one deals with this problem specifically, and in fact shows how analytic continuation can handle the problems of all the series presented in the Numberphile video).
This is not actually true. One is not obliged to assume that S is finite in order to come to this proof; only that it is definite. That is to say, we assume that S is a number distinct from other numbers. Before starting the proof, we implicitly acknowledge that it is possible S is a finite number, but it is similarly possible that S is an infinite number. A number need not be finite in order to be definite.
The symbol ∞ does not refer to a number. However, there are numbers which are infinite, and operations which are not defined for ∞ are defined for these latter.My Blog: http://oncreationism.blogspot.co.uk/
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Originally posted by The Pixie View PostJust to be clear, those web sites stated that certain operations could be performed on infinity, and others could not. They (and I) are certainly not just treating the infinity symbol as another number. My position here is that 4 times infinity is infinity, but infinity minus infinity is undefined.
Yes, you did. Three in fact, in post #26. I will give them again.
http://physicsbuzz.physicscentral.co...utely-not.html
http://skullsinthestars.com/2014/01/...ome-would-say/
http://www.slate.com/blogs/bad_astro...ng_result.html
There is a term there 4S. You and Leonhard have said that multiplying infinity by 4 is undefined (though I disagree). For this term 4S to have meaning, you are therefore obliged to assume it is not infinite. I would say that that is a bad assumption in this case.
Can you clarify? I have just realised this links to what you said in post #21, and I am not sure what you are talking about.
So, let's say I have two Hyperreal numbers, a and b. The easiest way to assign a value to Hyperreals is as a limit of a sequence of numbers. These sequences do not converge over the set of Real numbers, but they have a definite value for Hyperreals. In this case, I'm going to give the following equalities:
a = (2, 4, 6, 8, 10, 12, ..., 2n, ...)
b = (10, 20, 30, 40, 50, 60, ..., 10n, ...)
Both of these numbers are infinite, in that they are both larger than any finite number; however, both are also definite numbers, which is to say that they are not ∞. The operations which are not defined for ∞ are, in fact, defined for these Hyperreals. For example:
a-a=0
b-a=(8, 16, 24, 32, 40, 48, ..., 8n, ...)
5a=b
b/a=5
Over half a century ago, it was proved that the Hyperreal numbers are logically sound if (and only if) the Real numbers are logically sound. Therefore, as long as you are not claiming that numbers like √2 and π and e cannot exist, you have no basis for claiming that definite-but-infinite numbers cannot exist."[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
--Thomas Bradwardine, De Continuo (c. 1325)
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Originally posted by Boxing Pythagoras View PostWhen we say, "There exists some number S such that..." and you insist, "S=∞," you are indeed treating the infinity symbol as a number. As Leonhard and I have both objected, this is incorrect.
The correct way to approach it is to say there may be some number or it may be infinite. However, if you do that, then your mathematically trickery does not work.
All three of these links state that the sum of all Natural numbers does not converge, a statement with which Leonhard and I both agree. None of them says that , as you've claimed.
http://physicsbuzz.physicscentral.co...utely-not.html
Very top of the page, we see: "Correction: Does 1+2+3+4+ . . . =-1/12? Absolutely Not!"
Go to the second page, and he says: "Originally posted by Boxing Pythagoras View PostWhen we say, "There exists some number S such that..." and you insist, "S=∞," you are indeed treating the infinity symbol as a number. As Leonhard and I have both objected, this is incorrect.
The correct way to approach it is to say there may be some number or it may be infinite. However, if you do that, then your mathematically trickery does not work.
All three of these links state that the sum of all Natural numbers does not converge, a statement with which Leonhard and I both agree. None of them says that , as you've claimed.
http://physicsbuzz.physicscentral.co...utely-not.html
Very top of the page, we see: Correction: Does 1+2+3+4+ . . . =-1/12? Absolutely Not!
Go to the second page, and he says: "But any way you slice it - 1+2+3+4+ . . . = -1/12 + infinity"
http://skullsinthestars.com/2014/01/...ome-would-say/
"For this latter series, as we add more and more of the terms to the total it just gets bigger and bigger, approaching the infinite.
http://www.slate.com/blogs/bad_astro...ng_result.html
"In the method you learned in high school, the series 1+2+3+4+5+… doesn’t converge and tends to go to infinity."
Something important about the first and last links - they are correcting early claims. These are people who originally believed that it was -1/12, but were subsequently persuaded otherwise. They are, if you like, hostile witnesses. What do you think it would take to persuade them?
Again, this is not the case. For 4S to have meaning, you are obliged to assume that it is a number, not that it is a finite number. The problem with your assertion is that ∞ is not a number.
If S is infinity, then, according to you, 4S has no meaning.
In fact, this follows from what you say here. Infinity is not a number, and you have assumed that S is a number.
By the way, neither you nor Leonhard have yet produced any evidence that 4 multiplied by infinity is undefined. Can you not find any web pages that support your claim?
Do you not wonder why that might be?
The Hyperreals describe numbers with infinite and/or infinitesimal parts, in a similar manner as describes numbers with Complex parts.
It is a while since I have done group theory, but it occurs to me, after reading about hyperreals, that integers form a group under the operation of addition (this is well established, see here for example). A proper of such a group is closure, which means that the result of any addition must necesssarily also be an integer.But any way you slice it - 1+2+3+4+ . . . = -1/12 + infinity
http://skullsinthestars.com/2014/01/...ome-would-say/
"For this latter series, as we add more and more of the terms to the total it just gets bigger and bigger, approaching the infinite."
http://www.slate.com/blogs/bad_astro...ng_result.html
"In the method you learned in high school, the series 1+2+3+4+5+… doesn’t converge and tends to go to infinity. "
Something important about the first and last links - they are correcting early claims. These are people who originally believed that it was -1/12 and posted saying that, but were subsequently persuaded otherwise. They are, if you like, hostile witnesses. What do you think it would take to persuade them?
Again, this is not the case. For 4S to have meaning, you are obliged to assume that it is a number, not that it is a finite number. The problem with your assertion is that ∞ is not a number.
In fact, this follows from what you say here. Infinity is not a number, and you have assumed that S is a number.
By the way, neither you nor Leonhard have yet produced any evidence that 4 multiplied by infinity is undefined. Can you not find any web pages that support your claim? Do you not wonder why that might be?
The Hyperreals describe numbers with infinite and/or infinitesimal parts, in a similar manner as describes numbers with Complex parts.
It is a while since I have done group theory, but it occurs to me, after reading about hyperreals, that integers form a group under the operation of addition (this is well established, see here for example). A property of such a group is closure, which means that the result of any addition must necesssarily also be an integer. So not -1/12.My Blog: http://oncreationism.blogspot.co.uk/
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Originally posted by The Pixie View PostThen the error is in the claim that there exists some number S. You are at that point building the conclusion into the premise.
The correct way to approach it is to say there may be some number or it may be infinite. However, if you do that, then your mathematically trickery does not work.
For example, the simplest way to prove that there is no rational number equal to √2 is to assume that there does exist such a number, in simplest form (since any rational number can be expressed in simplest form). It is then trivial to prove that both the numerator and denominator of such a number would need to be even, contradicting our statement that the ratio is in simplest form. Thus, there is no rational number which is equal to √2.
There is no contradiction in the case of the Sum of All Natural Numbers. Therefore, there is no good reason to assume that the number does not exist-- especially when the algorithm produces results which are consistent and definite.
Hmm, well, let us take a look.
http://physicsbuzz.physicscentral.co...utely-not.html
Very top of the page, we see: Correction: Does 1+2+3+4+ . . . =-1/12? Absolutely Not!
Go to the second page, and he says: "But any way you slice it - 1+2+3+4+ . . . = -1/12 + infinity"
http://skullsinthestars.com/2014/01/...ome-would-say/
"For this latter series, as we add more and more of the terms to the total it just gets bigger and bigger, approaching the infinite.
http://www.slate.com/blogs/bad_astro...ng_result.html
"In the method you learned in high school, the series 1+2+3+4+5+… doesn’t converge and tends to go to infinity."
I did not say infinity was a number, I said S could be infinity (tends to infinity if you prefer).
If S is infinity, then, according to you, 4S has no meaning.
By the way, neither you nor Leonhard have yet produced any evidence that 4 multiplied by infinity is undefined. Can you not find any web pages that support your claim?
It is a while since I have done group theory, but it occurs to me, after reading about hyperreals, that integers form a group under the operation of addition (this is well established, see here for example). A proper of such a group is closure, which means that the result of any addition must necesssarily also be an integer."[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
--Thomas Bradwardine, De Continuo (c. 1325)
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