1 + 2 + 3 +... = -1/12

[The Pixie]

That's actually not true, in the least. If there does not exist such a number, then at some point, our calculation should have produced a contradiction. This never occurs, therefore our assumption that there does exist such a number is not in error. This is the basis of all Proofs by Contradiction.

There is a contradiction, because my calculation gives a result of infinity. Infinity does not equal -1/12.

So, you have precisely one non-mathematician who argues that . Forgive me if I prefer Ramanujan to Buzz Skyline, where math is concerned.

I counted three. Here are some more. Do please note that all of them are aware of the claim that S = -1/12; they are all discussing the supposed proof presented, and saying why it is wrong.

http://plus.maths.org/content/infinity-or-just-112

"This is why mathematicians say that the sum
\[ 1+2+3+4+ ... \]
diverges to infinity. Or, to put it more loosely, that the sum is equal to infinity. "

http://www.mathaware.org/mam/2014/ca.../infinity.html

"The equation implicitly declares that the infinite sum is equal to some number, a number that we name S. But mathematicians agree that the sum above is not, in fact, equal to any number. Not –1/12, and not anything else."

http://goodmath.scientopia.org/2014/...ad-astronomer/

"And there is the first huge, gaping, glaring problem with the video. They assert that the Cesaro sum of a series is equal to the series, which isn't true.
From there, they go on to start playing with the infinite series in sloppy algebraic ways, and using the Cesaro summation value in their infinite series algebra. This is, similarly, not a valid thing to do.
...
What makes this worse is that it's obvious. There is no mechanism in real numbers by which addition of positive numbers can roll over into negative. It doesn't matter that infinity is involved: you can't following a monotonically increasing trend, and wind up with something smaller than your starting point."

This page has links to numerous more pages debunking the -1/12 claim.

http://aperiodical.com/2014/01/an-in...nus-a-twelfth/

There are a lot of mathematicians out there who have seen the argument, and have chosen to reject it.

And please note that they do not say Ramanujan was wrong. To say you choose Ramanujan over Buzz SDkyline is a false dichotomy. The mathematians on those websites are right and so is Ramanujan. Ramanujan was talking about a different type of summation. By the way, several of these web sites also discuss the Casimir effect, and how the -1/12 result is applicable there.

As both Leonhard and I have agreed, the sequence does not converge.

Well there is your inconsistency.

You say it does not converge, and you also say it gets to -1/12.

Neither of these sources claims that

They say "approaching the infinite" and "doesn’t converge and tends to go to infinity". Are we playing semantic games now?

And they are adamant it is not -1/12.

S cannot be infinity, because S is a number and infinity is not a number. Now, it is possible that no such S exists, but that should have turned up a Proof by Contradiction. It did no such thing, in this case.

Prove then that S is a number. You calculation relies on that assumption.

If 4 times infinity has no definite meaning. That is not to say it has no meaning. Allow me to clarify: multiplying infinity by any constant does not produce a definite value, which is what I meant by "undefined," in this case.

And yet most websites on the subject state that while infinity minus infinity is undefined, infinity multipled by a positve number is infinity. Just Google "multiply infinity" and take a look. In fact, given you have failed to find any web site that supports your claim, I kind of suspect you already have. Think about it.

[Boxing Pythagoras]

There is a contradiction, because my calculation gives a result of infinity. Infinity does not equal -1/12.

Once again, your calculation does not give a result of infinity. Infinity is not a number, and therefore cannot be the result of any calculation.

I counted three. Here are some more. Do please note that all of them are aware of the claim that S = -1/12; they are all discussing the supposed proof presented, and saying why it is wrong.

Only one of them claimed that , while the other two simply claimed that the series is divergent (which is not at all the same thing as saying it "equals infinity"). All three disputed the notion that S is actually -1/12, but that is not the same as claiming S=∞, as you have.

http://plus.maths.org/content/infinity-or-just-112

"This is why mathematicians say that the sum
\[ 1+2+3+4+ ... \]
diverges to infinity. Or, to put it more loosely, that the sum is equal to infinity. "

Cool, now you have two non-mathematicians who argue that . I'm still going to side with Euler and Ramanujan on this one.

http://www.mathaware.org/mam/2014/ca.../infinity.html

"The equation implicitly declares that the infinite sum is equal to some number, a number that we name S. But mathematicians agree that the sum above is not, in fact, equal to any number. Not –1/12, and not anything else."

This gentleman is certainly not claiming that , and is quite explicitly claiming that S has no assignable value, at all.

http://goodmath.scientopia.org/2014/...ad-astronomer/

"And there is the first huge, gaping, glaring problem with the video. They assert that the Cesaro sum of a series is equal to the series, which isn't true.
From there, they go on to start playing with the infinite series in sloppy algebraic ways, and using the Cesaro summation value in their infinite series algebra. This is, similarly, not a valid thing to do.
...
What makes this worse is that it's obvious. There is no mechanism in real numbers by which addition of positive numbers can roll over into negative. It doesn't matter that infinity is involved: you can't following a monotonically increasing trend, and wind up with something smaller than your starting point."

As much as I usually love Phil Plait, he's actually quite wrong, here, and seems entirely ignorant that the claim that S=-1/12 is supported by more than just Cesaro summation. It is also supported by the Euler-Riemann Zeta function, by Ramanujan summation, and by Continual Analysis.

And please note that they do not say Ramanujan was wrong. To say you choose Ramanujan over Buzz SDkyline is a false dichotomy. The mathematians on those websites are right and so is Ramanujan. Ramanujan was talking about a different type of summation.

The article which you linked from David Berman explicitly claims that Ramanujan was wrong. Furthermore, anyone who claims that S does not equal -1/12 most certainly is claiming that Ramanujan is wrong, since Ramanujan's claim was that "1+2+3+4+...=-1/12."

Source: Ramanujan's 2nd Letter to GH Hardy, 27 February 1913

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.

© Copyright Original Source

Well there is your inconsistency.

You say it does not converge, and you also say it gets to -1/12.

I do not say that it "gets to" -1/12. I say that it equals -1/12. There is no succession of events, involved; no motion of any actors "getting" from one place to another. The series does not converge. Nonetheless, the series equals -1/12.

They say "approaching the infinite" and "doesn’t converge and tends to go to infinity". Are we playing semantic games now?

These are not semantics games, at all. "Doesn't converge" and "tends towards infinity" are in no way, shape, or form the same as saying "equals infinity."

Prove then that S is a number. You calculation relies on that assumption.

Once again, as no contradiction occurs, as the math leads us to a definite value, and as that definite value is most certainly a number, we have shown that S is a number. This requires no greater assumption than when you assume that 2+2 has a numerical solution.

And yet most websites on the subject state that while infinity minus infinity is undefined, infinity multipled by a positve number is infinity. Just Google "multiply infinity" and take a look. In fact, given you have failed to find any web site that supports your claim, I kind of suspect you already have. Think about it.

Not a single one of the websites which you listed claims that Infinity is a number. The only ones which claim that an expression can "equal infinity" are written by non-mathematicians.

Since you seem to attach an inordinate amount of importance to websites, here are a few which support this claim:
http://mathforum.org/library/drmath/view/62486.html
http://nrich.maths.org/2756
http://www.ditutor.com/limits/infinity.html

You may recognize that last one, as you have listed it yourself. Unfortunately, it seems that you ignored the section at the top when citing it (emphasis added):

Source: ditutor

Infinity (denoted by ∞) is a mathematical concept that refers to a quantity being boundless or having no end.

We must be clear that infinity is not a number.

We should note that these guidelines are not actual operations, but simply a resource to help solve limits.

© Copyright Original Source

[The Pixie]

Once again, your calculation does not give a result of infinity. Infinity is not a number, and therefore cannot be the result of any calculation.

Okay, technically yes, but tends to infinite can be the conclusion, and such a conclusion indicates the result is not a finite number.

Only one of them claimed that , while the other two simply claimed that the series is divergent (which is not at all the same thing as saying it "equals infinity"). All three disputed the notion that S is actually -1/12, but that is not the same as claiming S=∞, as you have.

Okay, you have convinced me. S tends to the infinite, rather than is equal to infinity.

Happy now?

So I have about a dozen web pages that agree with me that S is not -1/12, but in fact tends to infinity.

Cool, now you have two non-mathematicians who argue that . I'm still going to side with Euler and Ramanujan on this one.

That false dichotomy again.

No one is saying Euler and Ramanujan are wrong. We are all saying they are doing something different.

This gentleman is certainly not claiming that , and is quite explicitly claiming that S has no assignable value, at all.

So he to says it is not -1/12?

Will you accept it is not -1/12 then?

As much as I usually love Phil Plait, he's actually quite wrong, here, and seems entirely ignorant that the claim that S=-1/12 is supported by more than just Cesaro summation. It is also supported by the Euler-Riemann Zeta function, by Ramanujan summation, and by Continual Analysis.

Different types of summations, as several of the web pages I cited make clear.

I repeat, none of them are saying Ramanujan is wrong.

The article which you linked from David Berman explicitly claims that Ramanujan was wrong. Furthermore, anyone who claims that S does not equal -1/12 most certainly is claiming that Ramanujan is wrong, since Ramanujan's claim was that "1+2+3+4+...=-1/12."

I think you are misreading it:

"So where does the -1/12 come from? The wrong result actually appeared in the work of the famous Indian mathematician Srinivasa Ramanujan in 1913 (see this article for more information). But Ramanujan knew what he was doing and had a reason for writing it down. He had been working on what is called the Euler zeta function."

Source: Ramanujan's 2nd Letter to GH Hardy, 27 February 1913

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.

© Copyright Original Source

I do not say that it "gets to" -1/12. I say that it equals -1/12. There is no succession of events, involved; no motion of any actors "getting" from one place to another. The series does not converge. Nonetheless, the series equals -1/12.

These are not semantics games, at all. "Doesn't converge" and "tends towards infinity" are in no way, shape, or form the same as saying "equals infinity."

Once again, as no contradiction occurs, as the math leads us to a definite value, and as that definite value is most certainly a number, we have shown that S is a number. This requires no greater assumption than when you assume that 2+2 has a numerical solution.

The maths leads to that value because you are implicitly assuming S is finite, and you are very clear that maths breaks down if it is not. You are also very clear that it diverges...

Not a single one of the websites which you listed claims that Infinity is a number. The only ones which claim that an expression can "equal infinity" are written by non-mathematicians.

And nor am I. I have said right from the start that infinity minus infinity is not defined, and that was my reason for rejecting Leonhard's supposed proof.

Nevertheless, some operations on infinity are defined, and I provided a bunch of webpages that confirm that.

Since you seem to attach an inordinate amount of importance to websites, here are a few which support this claim:
http://mathforum.org/library/drmath/view/62486.html
http://nrich.maths.org/2756
http://www.ditutor.com/limits/infinity.html

Congratulation, you found a bunch of web sites that agree with us both.

Let me know when you find one that confirms your claim that 4 times infinity is not defined.

You may recognize that last one, as you have listed it yourself. Unfortunately, it seems that you ignored the section at the top when citing it (emphasis added):

Source: ditutor

Infinity (denoted by ∞) is a mathematical concept that refers to a quantity being boundless or having no end.

We must be clear that infinity is not a number.

We should note that these guidelines are not actual operations, but simply a resource to help solve limits.

© Copyright Original Source

Did you notice what it said for infinity minus infinity? "Indeterminate form"

And infinity multplied by a number? Infinity.

[klaus54]

Ramanujan summation is not the "usual" summation.

Is that the issue here? If so, it seems quite clear to me.

And I agree with the nomenclature that the sum of natural numbers diverges rather than is "equal to infinity." In fact I don't what the latter even means.

The divergent beeswax is how it's taught in Real Analysis, at least in my undergrad and graduate courses.

Interesting discussion!

K54

[Boxing Pythagoras]

Okay, technically yes, but tends to infinite can be the conclusion, and such a conclusion indicates the result is not a finite number.

Okay, you have convinced me. S tends to the infinite, rather than is equal to infinity.

Happy now?

Quite happy! This is a major difference, and one which eliminates your previous objections. Since S is not, in fact, infinity it is clear that we are not attempting to multiply ∞ by 4 or to subtract ∞ from ∞.

So I have about a dozen web pages that agree with me that S is not -1/12, but in fact tends to infinity.

I honestly do not understand why you are appealing to web pages-- especially when so many of the web pages to which you are appealing are not written by mathematicians. The fact that it is written somewhere on the Internet hardly makes it a convincing argument. Furthermore, Leon and I have both agreed-- wholeheartedly and several times-- that the series tends towards infinity. The fact that the partial sums tend toward infinity does not preclude the complete sum from having a value.

So he to says it is not -1/12?

Will you accept it is not -1/12 then?

Actually, he says, "even though S is a divergent series, if it were to have a value, –1/12 would be a sensible one." I disagree with his assertion that the sum cannot be assigned any value, particularly because his only defense of that claim is the fact that S is a divergent series.

Different types of summations, as several of the web pages I cited make clear.

This does not make them any less summations.

That false dichotomy again.

No one is saying Euler and Ramanujan are wrong. We are all saying they are doing something different.

I repeat, none of them are saying Ramanujan is wrong.

I think you are misreading it:

"So where does the -1/12 come from? The wrong result actually appeared in the work of the famous Indian mathematician Srinivasa Ramanujan in 1913 (see this article for more information). But Ramanujan knew what he was doing and had a reason for writing it down. He had been working on what is called the Euler zeta function."

Ramanujan stated that "1+2+3+4+...=-1/12." If you say that 1+2+3+4+... does not equal -1/12, then you are saying Ramanujan was wrong. You cannot have your cake and eat it, too.

The maths leads to that value because you are implicitly assuming S is finite, and you are very clear that maths breaks down if it is not. You are also very clear that it diverges...

Once again, there is no assumption that S is finite. There is only the assumption that S is definite. As I have demonstrated, definite numbers need not be finite numbers.

And nor am I. I have said right from the start that infinity minus infinity is not defined, and that was my reason for rejecting Leonhard's supposed proof.

Nevertheless, some operations on infinity are defined, and I provided a bunch of webpages that confirm that.

Congratulation, you found a bunch of web sites that agree with us both.

Let me know when you find one that confirms your claim that 4 times infinity is not defined.

Did you notice what it said for infinity minus infinity? "Indeterminate form"

And infinity multplied by a number? Infinity.

Once again, you really ought to read that disclaimer at the top of the DITUTOR page.

Source: ditutor

Infinity (denoted by ∞) is a mathematical concept that refers to a quantity being boundless or having no end.

We must be clear that infinity is not a number.

We should note that these guidelines are not actual operations, but simply a resource to help solve limits.

© Copyright Original Source

When he says that "these guidelines are not actual operations," he is referring to the fact that these operations are not defined for ∞. Similarly, the page which I cited from Dr. Wallace also explicitly states precisely what I have been saying, which is that numerical operations are not defined for things which are not numbers.

Source: Ask Dr. Math

The very sentence "1/infinity = 0" has no meaning. Why? Because "infinity" is a concept, NOT a number. It is a concept that means "limitlessness." As such, it cannot be used with any mathematical operators. The symbols of +, -, x, and / are arithmetic operators, and we can only use them for numbers.

To write 1/infinity and mean "1 divided by infinity" doesn't make any sense. 1 cannot be divided by a concept. It can only be divided by a number. Similarly, "infinity + 1" or "2 times infinity" are also meaningless.

© Copyright Original Source

"Infinity multiplied by a number" is a nonsensical statement. It is absolutely meaningless. Furthermore, since we have now agreed that S does not equal ∞, your objections against our performing certain operations on S have fallen through. It makes no sense to object to 4S-S based on the fact that ∞-∞ is indeterminate, since we both agree that S≠∞.

[The Pixie]

Consider three series:

S =  Sum(i)    = 1 + 2 + 3 + 4 + 5 ...
S1 = Sum(i^0)  = 1 + 1 + 1 + 1 + 2 ...
S2 = Sum(2i^0) = 2 + 2 + 2 + 2 + 2 ...

We start by getting a finite value of S1 from S

S = 1 + 2 + 3 + 4 + 5 ...
  = (0+1) + (1+1) + (2+1) + (3+1) + (4+1) ...
  = 0 + (1+2+3+4...) + (1+1+1+1+1...)
  = 0 + S + S1
=>
S1 = 0

We can also determine a finite value of S2 from S

S = 1 + 2 + 3 + 4 + 5 ...
  = (-1+2) + (0+2) + (1+2) + (2+2) + (3+2) + (4+2) ...
  = -1+0+1+2+3+4... + 2+2+2+2+2...
  = -1 + 0 + S + S2
=>
S2 = 1                  ... result 1

But wait

S2 = 2 + 2 + 2 + 2 + 2 ...
   = (1+1) + (1+1) + (1+1) + (1+1) + (1+1)...
   = (1+1+1+1+1...) + (1+1+1+1+1...)
   = S1 + S1
   = 0 + 0
S2 = 0                  ... result 2

Bringing all that together:

S2 = 1                  ... from result 1
S2 = 0                  ... from result 2
=>
1 = 0

I invite Boxing Pythagoras and Leonhard to point out the flaw in this reasoning. I think the flaw here is the same as in the argument Leonhard used in the OP. If you guys are convinced that his argument is sound, you need to find another flaw or accept that one is equal to zero!

[Boxing Pythagoras]

I invite Boxing Pythagoras and Leonhard to point out the flaw in this reasoning. I think the flaw here is the same as in the argument Leonhard used in the OP. If you guys are convinced that his argument is sound, you need to find another flaw or accept that one is equal to zero!

Ahhhh, now this is a much better argument than before. Yes, it is quite dangerous to manipulate divergent infinite sums, and this can lead to all sorts of nonsense when done without proper guidance. The proof from Leonhard's OP is not very rigorous. However, it is far easier for the average person to understand than other methods, and it produces results consistent with rigorous proofs, so it can be a very useful tool for demonstrating this extremely counterintuitive result. Still, a more rigorous proof is definitely preferable when it is shown that these purely arithmetic manipulations have shortcomings.

There are a few more rigorous proofs that the sum of all Natural numbers equals -1/12. For example, we could extend the Euler-Riemann zeta function using continual analysis to show that, in general, , where B represents the Bernoulli numbers. Then, by simply plugging in s=1, we can find that:


However, it is entirely possible to prove this result without resorting to complex continual analysis, and to do so in a manner which resolves some apparent inconsistencies with more intuitive arithmetic derivations, such as those we have explored in this thread. However, rather than attempting to reinvent the wheel by posting that whole discussion, here, I'll simply point you over to this article by Fields medalist, Dr. Terrence Tao. Incidentally, he won the Fields medal for his work in additive number theory.

I understand how wildly counterintuitive it is to think that the sum of a divergent, infinite series of positive integers could be equal to a negative rational number. However, I'll point out that it was once just as counterintuitive for people to think that the sum of any infinite series-- even a convergent one-- could actually be assigned a definite, finite value. During the 17th Century, there were vehement and heated arguments between mathematicians, philosophers, and theologians on the subject, with personalities as diverse as Jacob Bidermann (head of the Revisors General of the Society of Jesus) and Thomas Hobbes (noted philosopher and accused atheist) arguing that the adoption of such mathematics would result in nothing less than the downfall of all human reason.

Not only has this not led to the downfall of human reason, but such mathematics has proved entirely consistent and completely applicable to the real world. In exactly the same way, the mathematics which leads us to believe that the sum of all Natural numbers equals -1/12 has proved to be entirely consistent and completely applicable to the real world.

[The Pixie]

... Yes, it is quite dangerous to manipulate divergent infinite sums, and this can lead to all sorts of nonsense when done without proper guidance. ...

I wish I had said that. Oh wait, I did (okay, I called it infinity, but the point is the same; you are pretending S has a definite value when it does not):

Post #17
You cannot do that. The two partial sums are infinite. What is infinity minus infinity? Zero, fourteen, infinity, negative infinity? All these and none of them. It is undefined.

Post #20
Step 4 you have 4S on the left. But S is infinity, so 4S is four times infinity, which is infinity!
When you go to step 5, you subtract infinity from both sides, then divide through by 3. Somehow four times infinity minus infinity is three times infinity!

Post #33
But you are making the assumption that S is not infinite to prove that it is not infinite. Does that not strike you as somewhat circular?

Post #35
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.

Post #37
But you are trying to prove S is finite, and to do that, you are obliged to assume it is finite!

Post #41:
So therefore the term 4S is not properly defined if S in infinite.
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.

Post #42
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.

Post #44
Then the error is in the claim that there exists some number S. You are at that point building the conclusion into the premise.

Post #44
Then 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.

Post #48
I have said right from the start that infinity minus infinity is not defined, and that was my reason for rejecting Leonhard's supposed proof.

Oh, and let us not forget all the websites that said the same thing, but nevertheless you rubbished.

What Leonhard did in the OP is mathematical trickery. To say "it is far easier for the average person to understand than other methods" misses the point that it is teaching peoople bad practices. That he ended up with a result that appears to have some meaning does not make it any less trickery. His calculation is no more valid than mine, where I "proved" the one equals zero.

I will note that you have yet to point to any step in my "proof" that is not valid. As far as I can see it all follows logically from the claim that S has a finite value. If you cannot find any flaw, then you have to accept that one equals zero. Or, as a last resort, accept the initial premise is wrong.

[Boxing Pythagoras]

I wish I had said that. Oh wait, I did (okay, I called it infinity, but the point is the same; you are pretending S has a definite value when it does not):

The claim "S=∞" is not at all the same as "the arbitrary manipulation of arithmetic sequences of infinite divergent series lacks rigor." Nor does the latter, true statement imply that S has no definite value. The fact of the matter is that S does have a definite value, and while the method given by the OP is not rigorous, there are ways to prove this which are rigorous and which do provide a true and consistent framework for establishing the definite value of a sum of an infinite, divergent series.

What Leonhard did in the OP is mathematical trickery. To say "it is far easier for the average person to understand than other methods" misses the point that it is teaching peoople bad practices. That he ended up with a result that appears to have some meaning does not make it any less trickery. His calculation is no more valid than mine, where I "proved" the one equals zero.

I can agree with this, to a point. However, so long as one is aware of the shortcomings and limitations of a particular algorithm, that algorithm can still be quite useful.

I will note that you have yet to point to any step in my "proof" that is not valid. As far as I can see it all follows logically from the claim that S has a finite value. If you cannot find any flaw, then you have to accept that one equals zero. Or, as a last resort, accept the initial premise is wrong.

Yes, purely arithmetic manipulations are inconsistent. This would be an example of a Proof by Contradiction which shows that such methods are not generally applicable and, at best, can only be applied in special cases where the outcome is supported by more rigorous work.

Take, as an example, ancient Greek mathematics. It had been well established and irrefutably proved that, in any right angled triangle, the sum of the squares on the legs enclosing the right angle is equal to the square on the side subtending the right angle. This is now known as the Pythagorean Theorem. Well, along comes Hippasus, who uses this well-established fact to show that there cannot possibly exist any ratio which describes the length of a square's diagonal in terms of the length of its side-- which he showed through a fairly simple and straightforward Proof by Contradiction. Purportedly, Hippasus' discovery had so maddened his Pythagorean brothers that they through him over the side of a ship and let him drown. However, there were a few possible implications of Hippasus' work:

1. The assumed correctness of the Pythagorean Theorem was in error.
2. There really does not exist any such number, meaning that the diagonal of a square is actually an illusory and nonexistent thing.
3. There does exist some definite value to describe the number, but it requires new mathematics.

Numbers (1) and (2) were unthinkable, as it would require a person to basically discard everything which had yet been discovered about geometry. So, the only possibly correct answer was (3)-- which is now the indisputably accepted answer. However, more than 2000 years would pass between the acceptance of this incongruence and a rigorous method for defining such irrational numbers.

Here's another example. It is easy to show the general case of the Difference of Squares rule, which is to say . Now, let's look at one of my favorite numbers: 5. We can very easily show that . However, there exists no Real number which corresponds to √(-1). So, the Difference of Squares rule is not generally applicable to all Real number cases. Does that mean that we should abandon using the Difference of Squares rule, entirely, when we are dealing with Real numbers? Certainly not! So long as we are aware of its limitations, it can be used to great effect.

The sum of all Natural numbers is a similar problem. Classical methods of summation cannot deal with infinite sums, at all, and basic calculus only gives us a rigorous method for summing infinite convergent series. So, either no such sums exist (as you are now asserting, and as a few of the pages which you cited claim) or else we need to find some other mathematics to evaluate such sums (as Cesaro, Abel, Euler, Ramanujan, Tao, and numerous other exceptional mathematicians have done).

So, yes, the algorithm for arriving at the value shown in the OP is not rigorous, nor is it generally applicable. However, that does not mean that the result at which it arrives is in error.

[Leonhard]

Consider three series:

S =  Sum(i)    = 1 + 2 + 3 + 4 + 5 ...
S1 = Sum(i^0)  = 1 + 1 + 1 + 1 + 2 ...
S2 = Sum(2i^0) = 2 + 2 + 2 + 2 + 2 ...

We start by getting a finite value of S1 from S

S = 1 + 2 + 3 + 4 + 5 ...
  = (0+1) + (1+1) + (2+1) + (3+1) + (4+1) ...
  = 0 + (1+2+3+4...) + (1+1+1+1+1...)
  = 0 + S + S1
=>
S1 = 0

We can also determine a finite value of S2 from S

S = 1 + 2 + 3 + 4 + 5 ...
  = (-1+2) + (0+2) + (1+2) + (2+2) + (3+2) + (4+2) ...
  = -1+0+1+2+3+4... + 2+2+2+2+2...
  = -1 + 0 + S + S2
=>
S2 = 1                  ... result 1

But wait

S2 = 2 + 2 + 2 + 2 + 2 ...
   = (1+1) + (1+1) + (1+1) + (1+1) + (1+1)...
   = (1+1+1+1+1...) + (1+1+1+1+1...)
   = S1 + S1
   = 0 + 0
S2 = 0                  ... result 2

Bringing all that together:

S2 = 1                  ... from result 1
S2 = 0                  ... from result 2
=>
1 = 0

AAARRRRGHH... MY EYES!!!!

The Pixie... go to this Online LaTeX Equation Editor, it will be your trusty friend from now on. Write the equation there, if you look down you'll find a link (choose URL Encoded). If you post that link between [img ][/img ] tags (no need to download the image), you'll display a beautiful equation instead.

But let me do it for you. I'll rewrite your post: (speaking as The Pixie)

"Consider the three series:







We start by getting a finite value of S1 from S



We can also determine a finite value of S2 from S



But wait



Bringing all that together:

[The Pixie]

Thanks Leonhard for the attractive formatting!

The claim "S=∞" is not at all the same as "the arbitrary manipulation of arithmetic sequences of infinite divergent series lacks rigor." Nor does the latter, true statement imply that S has no definite value. The fact of the matter is that S does have a definite value, and while the method given by the OP is not rigorous, there are ways to prove this which are rigorous and which do provide a true and consistent framework for establishing the definite value of a sum of an infinite, divergent series.

So what you are saying is that my mathematical language was lax (which I accept). However, in my first post on this thread I said:

"You cannot do that. The two partial sums are infinite. What is infinity minus infinity? Zero, fourteen, infinity, negative infinity? All these and none of them. It is undefined."

It was the methodology of the OP to which I was objecting. I stand by that objection. The supposed proof in the OP is flawed. I see nothing here that makes me think otherwise.

I can agree with this, to a point. However, so long as one is aware of the shortcomings and limitations of a particular algorithm, that algorithm can still be quite useful.

What shortcomings? Why so vague?

The reason is that the shortcomings of my proof are also there in the OP. If you get specific about mine, it will be clear that is also true of the OP.

Yes, purely arithmetic manipulations are inconsistent. This would be an example of a Proof by Contradiction which shows that such methods are not generally applicable and, at best, can only be applied in special cases where the outcome is supported by more rigorous work.

What does this mean? Are you talking about arithmetic manipulations of diverging series?

If so, then yes, such arithmetic manipulations are indeed inconsistent. Which means the supposed proof in the OP is flawed. Which is my point.

Take, as an example, ancient Greek mathematics. It had been well established and irrefutably proved that, in any right angled triangle, the sum of the squares on the legs enclosing the right angle is equal to the square on the side subtending the right angle. This is now known as the Pythagorean Theorem. Well, along comes Hippasus, who uses this well-established fact to show that there cannot possibly exist any ratio which describes the length of a square's diagonal in terms of the length of its side-- which he showed through a fairly simple and straightforward Proof by Contradiction. Purportedly, Hippasus' discovery had so maddened his Pythagorean brothers that they through him over the side of a ship and let him drown. However, there were a few possible implications of Hippasus' work:

1. The assumed correctness of the Pythagorean Theorem was in error.
2. There really does not exist any such number, meaning that the diagonal of a square is actually an illusory and nonexistent thing.
3. There does exist some definite value to describe the number, but it requires new mathematics.

Numbers (1) and (2) were unthinkable, as it would require a person to basically discard everything which had yet been discovered about geometry. So, the only possibly correct answer was (3)-- which is now the indisputably accepted answer. However, more than 2000 years would pass between the acceptance of this incongruence and a rigorous method for defining such irrational numbers.

What do you think this proves? Do you think Hippasus' Proof by Contradiction is valid?

Here's another example. It is easy to show the general case of the Difference of Squares rule, which is to say . Now, let's look at one of my favorite numbers: 5. We can very easily show that . However, there exists no Real number which corresponds to √(-1). So, the Difference of Squares rule is not generally applicable to all Real number cases. Does that mean that we should abandon using the Difference of Squares rule, entirely, when we are dealing with Real numbers? Certainly not! So long as we are aware of its limitations, it can be used to great effect.

Right. And when we consider divergent series, we know that arithmetic manipulations are not appropriate. Thus is was wrong to use them in the OP. Which is the essence of what I said back in post #17.

The sum of all Natural numbers is a similar problem. Classical methods of summation cannot deal with infinite sums, at all, and basic calculus only gives us a rigorous method for summing infinite convergent series. So, either no such sums exist (as you are now asserting, and as a few of the pages which you cited claim) or else we need to find some other mathematics to evaluate such sums (as Cesaro, Abel, Euler, Ramanujan, Tao, and numerous other exceptional mathematicians have done).

So, yes, the algorithm for arriving at the value shown in the OP is not rigorous, nor is it generally applicable. However, that does not mean that the result at which it arrives is in error.

But it also does not mean the result is correct either, and it certainly does mean that the supposed proof in the OP (and the Numberphile video you presented) is flawed. It works by mathematically trickery using operations you have finally admitted are not consistent in this situation.

Let us look back at post #21:

Step 4 you have 4S on the left. But S is infinity, so 4S is four times infinity, which is infinity!

Infinity is not a number. So the statement "S is infinity" is not meaningful, in this context. Step 4 is just a simple application of the Distributive Property of Multiplication, and it's fairly obviously true.

Here you are claiming that 4S is valid; that it is consistent to multiply a divergent series by four.

You in post #30, with regards to S:

We can multiply a well-defined number by 4, and we can subtract from a well-defined number.

Do you still maintain that we can do that with a divergent sum?

[Leonhard]

The Pixie, I think I've already admitted that the proof I gave wasn't entirely stringent, it was meant more as a pedagogic means of deriving the answer. So if all you're trying to is make me say that again, you're kinda beating a dead horse.

I don't think your disproof works because I never engaged in the kind of manipulation of the series that you do. The types of partial sums actually change in between some of your lines, but that didn't take place in my derivation. And so for this simple reason, you don't have a true counterexample, and I don't think you'll find one.

[The Pixie]

The Pixie, I think I've already admitted that the proof I gave wasn't entirely stringent, it was meant more as a pedagogic means of deriving the answer. So if all you're trying to is make me say that again, you're kinda beating a dead horse.

If Boxing Pythagoras can admit to, then my work here is done.

I don't think your disproof works because I never engaged in the kind of manipulation of the series that you do. The types of partial sums actually change in between some of your lines, but that didn't take place in my derivation.

Can you explain that?

[Leonhard]

Can you explain that?

Certainly, as I stated very early in the discussion, results like -1/12, are to be seen primarily as property of the partial sums, so if you make changes to terms of the partial sum, you change this property. And if you don't keep track of that, you can easily derive contradictions.

Take the first one



You then write



And so far its good, and if you had written it



Things would have been fine, but you decide instead to shift the entire partial sum, of one of the sums, one index to the left.



And this is not quite the same, hence you manage to derive a contradiction which doesn't surprise me.

You can always shift indexes for finite sums and have it come out meaningful, and for all series where the partial sum of all terms up to n converges for n going towards infinity, however in the case of divergent series you can get different answers by shifting the sums like this.

However since I didn't do that, your counterexample doesn't work.

[Boxing Pythagoras]

So what you are saying is that my mathematical language was lax (which I accept). However, in my first post on this thread I said:

"You cannot do that. The two partial sums are infinite. What is infinity minus infinity? Zero, fourteen, infinity, negative infinity? All these and none of them. It is undefined."

It was the methodology of the OP to which I was objecting. I stand by that objection. The supposed proof in the OP is flawed. I see nothing here that makes me think otherwise.

If you had said, "Arithmetic manipulations of the sums of infinite, divergent series are not rigorous and inconsistent," I would have had no objection. Instead, you claimed that the algorithm in the OP requires one to subtract infinity from infinity, which is not true.

What shortcomings? Why so vague?

The reason is that the shortcomings of my proof are also there in the OP. If you get specific about mine, it will be clear that is also true of the OP.

I thought I was rather clear. Purely arithmetic manipulations of sums of infinite, divergent series are not rigorous and inconsistent. Yes, this does also apply to the algorithm in the OP. However, the algorithm in the OP is supported by other proofs-- proofs which are rigorous and consistent.

What does this mean? Are you talking about arithmetic manipulations of diverging series?

If so, then yes, such arithmetic manipulations are indeed inconsistent. Which means the supposed proof in the OP is flawed. Which is my point.

A point with which both Leonhard and I have agreed! Our objection was not to the idea that such arithmetic manipulations are inconsistent. Our objection was to your claim that the flaw in the OP had to do with attempting to perform operations on infinity, which it was not doing.

What do you think this proves? Do you think Hippasus' Proof by Contradiction is valid?

It demonstrates that an inconsistency in a particular algorithm does not invalidate results which are supported by other, more rigorous mathematics.

Right. And when we consider divergent series, we know that arithmetic manipulations are not appropriate. Thus is was wrong to use them in the OP. Which is the essence of what I said back in post #17.

That's actually not at all what you said, in post #17. What you said in post #17 was that the issue was an attempt to perform arithmetic operations on infinity, which was not the case.

But it also does not mean the result is correct either, and it certainly does mean that the supposed proof in the OP (and the Numberphile video you presented) is flawed. It works by mathematically trickery using operations you have finally admitted are not consistent in this situation.

If you had based your early objections on the lack of rigor, I would have agreed with you and moved on to more rigorous proofs at that point. Instead you focused your objections on incoherent arguments about multiplying and subtracting infinity.

Let us look back at post #21:

Here you are claiming that 4S is valid; that it is consistent to multiply a divergent series by four.

You in post #30, with regards to S:

Do you still maintain that we can do that with a divergent sum?

There is no "divergent sum." There is a sum of a divergent series. So long as the sum of the series is a well-defined number, it does not matter whether that series diverges or converges. You can perform mathematical operations on well-defined numbers. So, yes, I maintain my position from post #21 in which I stated that you were wrong to equate S with infinity, and I maintain my position from post #30 that we can multiply a well-defined number by 4 and we can subtract from a well-defined number. At no point has it been shown that there is any error in considering S to be a well-defined number.