Sum of all positive integers

In the YouTube clip
YouTube
"ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12"
they give a proof that the sum of all positive integers is negativ one twelfth !!!

I've seen the clip, again and again, over and over, and I become more and more frustrated!
It just cannot be true, it must be plus infinite, it must be positive, it just cannot be -1/12, a negative fraction!

Where is the error - if there is an error - where is it?

Originally posted by FabianFnas
In the YouTube clip
https://www.youtube.com/watch?v=w-I6XTVZXww
"ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12"
they give a proof that the sum of all positive integers is negativ one twelfth !!!

I've seen the clip, again and again, over and over, and I become more and more frustrated!
It just cannot be true, it must be plus infinite, it must be posi ...[text shortened]... annot be -1/12, a negative fraction!

Where is the error - if there is an error - where is it?

This has actually been covered several times in this forum.
In general:
1. Infinite sums are defined differently from ordinary sums. (and there is more than one way to do it).
2. Always be careful with infinities.
3. Infinity isn't a real number and therefore is not the correct answer either.
4. Other than the definition used to get -1/12 (which has proved useful in physics), a better way to handle it is to talk of limits, in which case the partial sums tend to infinity (which is what you would expect).

Originally posted by twhitehead
This has actually been covered several times in this forum.
In general:
1. Infinite sums are defined differently from ordinary sums. (and there is more than one way to do it).
2. Always be careful with infinities.
3. Infinity isn't a real number and therefore is not the correct answer either.
4. Other than the definition used to get -1/12 (which has ...[text shortened]... alk of limits, in which case the partial sums tend to infinity (which is what you would expect).

I've been gone for a while and have missed the other threads, I'm afraid. I have to go and find them.

Infinite sums - as they are contra-intuitive - can they be used in some serious branches of reality?

Originally posted by FabianFnas
I've been gone for a while and have missed the other threads, I'm afraid. I have to go and find them.

Infinite sums - as they are contra-intuitive - can they be used in some serious branches of reality?

https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#Physics

https://en.wikipedia.org/wiki/Zeta_function_regularization

Originally posted by twhitehead
https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#Physics

https://en.wikipedia.org/wiki/Zeta_function_regularization

Thank you for the links, twhitehead, now I have some to digest.

FabianFnas

NO NO NO don't listen to them! It is a MYTH!
They made fatal logical errors in the maths!

In another thread a very long time ago, After I mulled over some of the links, I found one that clearly implies that the infinite series:

1 + 2 + 3 + 4 …

cannot possibly equal -1/12 !

http://en.wikipedia.org/wiki/Divergent_series
“...In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series must approach zero.
Thus any series in which the individual terms do not approach zero diverges. ...”

The above statement is accurate and ALWAYS TRUE. Thus the infinite series:

1 + 2 + 3 + 4 …


cannot possibly converge to -1/12 or anything finite since it is clear that the individual terms do not approach zero.

Since the above statement in that link is accurate and always true, the same goes for the infinite series

1 – 1 + 1 – 1 + 1...

but, in this case, it doesn't equal infinity but rather simply doesn't have a sum!

Thus the whole premise on that video link
YouTube
that claims to prove that the infinite sum 1 + 2 + 3 + 4 … equals -1/12 must be WRONG (and is all a bit of nonsense ) because that video link says the infinite series 1 – 1 + 1 – 1 + 1... converges on ½ while the http://en.wikipedia.org/wiki/Divergent_series link CORRECTLY implies it cannot converge because the individual terms don't tend to zero.

Yes, of course. But...

Say that A leads to B, B leads to C, ... Y leads to Z. That means that A leads to Z, right?
If A is correct but Z is not correct, then there is a fault somewhere in the chain.

I say that if A is correct because S = 1+2+3+4+... and Z is wrong because intuitively S cannot possibly be = -1/12. A sum of integers cannot be anything but integers, and a sum of positive integers must be positive. Simple as that.

But where in the video do they propose something, from B to Y, that violates math? I see none, but there must be somewhere.

Could it be that 1-1+1-1+... is in fact not equal to 1/2 ? Well, they prove it in another clip with the same methods that this actually is true. So, can it be this?

Originally posted by FabianFnas
A sum of integers cannot be anything but integers,

And that is where you go wrong.
A sum of an infinite number of integers is NOT an integer.

Originally posted by twhitehead
And that is where you go wrong.
A sum of an infinite number of integers is NOT an integer.

When it is about a finite numbers of integers, then the sum is also a integer. So says common sense.
When it is about an in-finite number of integers, then the common sense is all wrong.

No, I don't get it, but I'm not supposed to get it either.
But I will still use it as a good party-trick though.

Originally posted by FabianFnas

Say that A leads to B, B leads to C, ... Y leads to Z. That means that A leads to Z, right?

Correct, but that's not the problem here; inference correct.The problem here is that 'A' (the premise) is false.

Could it be that 1-1+1-1+... is in fact not equal to 1/2 ?

BINGO! You got it!
And THAT is the faulty premise 'A' I am referring to.
The sum 1-1+1-1+... does not equal 1/2 but rather is undefined meaning it doesn't have a limit as it tends to infinity and the infinite series in this case has no sum!
Maths is often anti-intuitive but this is one of the cases where intuition just happens to be right and somebody's maths is what is wrong here.

Well, they prove it in another clip with the same methods that this actually is true.

I haven't seen that said proof but I assert any such said proof must be in error. But I would still like to see this said proof.

Originally posted by humy
Correct, but that's not the problem here; inference correct.The problem here is that 'A' (the premise) is false.

Could it be that 1-1+1-1+... is in fact not equal to 1/2 ?

BINGO! You got it!
And THAT is the faulty premise 'A' I am referring to.

Well, they prove it in another clip with the same methods that this actually is true.

I haven't seen that said proof but I assert any such said proof must be in error.

If you start with something that is true (A) and ends up with something that defy common sense (Z), then the error is somewhere in between.

What if (Z) is correct even if it's not common sense? Math is not always common sense for the general public.

The sum of all positive integers cannot both be plus infinite and -1/12 at the same time. Something is fishy. The interesting question is - what?

Originally posted by FabianFnas

What if (Z) is correct even if it's not common sense?

In this case it has nothing to do with common sense even though by irrelevant pure coincidence it happens to conform to common sense. The reason why 'A' (the premise) is wrong is not because of common sense saying it is wrong (even though it does) but rather because it can be logically deduced to be wrong. To start to see how it can be deduced to be wrong, read;

http://en.wikipedia.org/wiki/Divergent_series
“...If a series converges, the individual terms of the series must approach zero.
Thus any series in which the individual terms do not approach zero diverges. ...”

I assert that the above assertion is ALWAYS true i.e. they got that maths law exactly correctly stated there. And you should be able to see that if that above assertion is correct then the infinite series 1 - 1 + 1 ... does not converge thus it cannot equal 1/2.

Actually that above assertion in that link directly contradicts 1 + 2 + 3 ... converging to any number even without considering the fact it also contradicts the premise that same way.

This is a (relatively rare?) case of common sense just by happy 🙂 pure coincidence happening to be right.

Originally posted by humy
In this case it has nothing to do with common sense even though by irrelevant pure coincidence it happens to conform to common sense. The reason why 'A' (the premise) is wrong is not because of common sense saying it is wrong (even though it does) but rather because it can be logically deduced to be wrong. To start to see how it can be deduced to be wrong, read ...[text shortened]... a (relatively rare?) case of common sense just by happy pure coincidence happening to be right.

In the original Youtube clip they said that 1-1+1-1+... can be either plus one or zero. So 'we set it to be the average = 1/2'. This is BS if anyone ask me. One cannot just reason what the sum is. And this result he uses in the proof of sum of positive integers. Very sloppy, indeed it is.

But in this video YouTube he actually make a proof, in the same kind of reasoning as the other sum proof.

But even here I don't know where he is off track. Where is the error in his reasoning? It must be there somewhere! Where is the flaw?

Originally posted by FabianFnas
In the original Youtube clip they said that 1-1+1-1+... can be either plus one or zero. So 'we set it to be the average = 1/2'. This is BS if anyone ask me. One cannot just reason what the sum is. And this result he uses in the proof of sum of positive integers. Very sloppy, indeed it is.

But in this video https://www.youtube.com/watch?v=PCu_BNNI5x4 he ...[text shortened]... s off track. Where is the error in his reasoning? It must be there somewhere! Where is the flaw?

YouTube

I think that said 'proof' the infinite sum of that infinite series adding to 1/2 i.e.
1 - 1 + 1 - 1 + 1 - 1 ... = 1/2
is erroneous.

Lets go through his reasoning one step at a time;

He first says the infinite series;
1 - 1 + 1 - 1 + 1 - 1 ...
can be rewritten with an infinite series of added brackets as;
(1 - 1) + (1 - 1) + (1 - 1) ...
with what is in the brackets canceling down to 0 thus;
(0) + (0) + (0) ... = 0
and so he says that equals 0 'thus' we have;
1 - 1 + 1 - 1 + 1 - 1 ... = 0
But then he says that infinite series can be rewritten with an infinite series of added brackets as;
1 + (- 1 + 1)+ (- 1 + 1) +(- 1 ...
with what is in the brackets canceling down to 0 thus;
1 + (0) + (0) + (0) ... = 1 + 0 = 1
and so he says that equals 1 'thus' we have;
1 - 1 + 1 - 1 + 1 - 1 ... = 1
OK, already I think we have a big problem! For we now have that very SAME infinite series apparently equaling two different unequal numbers depending on how you completely arbitrary place the brackets. So you can now write the infinite series on both sides of the equal sign thus;
1 - 1 + 1 - 1 + 1 - 1 ... = 1 - 1 + 1 - 1 + 1 - 1 ...
which can be arbitrarily rewritten with an infinite series of added brackets differently on either side of that equals sign as;
(1 - 1) + (1 - 1) + (1 - 1) ... = 1 + (- 1 + 1)+ (- 1 + 1) +(- 1 ...
thus, using the previous results, that simplifies to;
0 = 1 Contradiction! 0 does not equal 1 !
I think this contradiction should tell us that the sum of that infinite series is not a definable number!

OK, but lets ignore that contradiction (like he did) and continue regardless!

He then says let;
S = 1 - 1 + 1 - 1 + 1 - 1 ...
This is implicitly assuming that you can validly write that as S which in turn implicitly assuming that that infinite series is a definable number, which it isn't.
he then posses the question what is 1 - S equal to?
He then writes this out as;
1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ... )
But then says that of you remove the brackets you must reverse the signs of the 1s in the brackets thus we get;
1 - S = 1 - 1 - 1 + 1 - 1 + 1 - 1 ...
and then says the RHS is the SAME infinite series we started with 'thus' we have;
1 - S = 1 - 1 - 1 + 1 - 1 + 1 - 1 ... = S
and that implies 1 - S = S which implies 1 = 2S which implies S = 1/2.
BUT, and here is the logical error, it being the SAME infinite series we started with doesn't mean it has the SAME value we started with! That is because we had previously given the results of BOTH;
1 - 1 + 1 - 1 + 1 - 1 ... = 0
AND
1 - 1 + 1 - 1 + 1 - 1 ... = 1
Thus using those two results you could just as easily write BOTH;
1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ... ) = 1 - (0) = 1
AND
1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ... ) = 1 - (1) = 0
So now we have yet additional contradictions of;
1 - 1 + 1 - 1 + 1 - 1 ... = 0
AND
1 - 1 + 1 - 1 + 1 - 1 ... = 1
AND
1 - 1 + 1 - 1 + 1 - 1 ... = 1/2
ALL three of the above contradict each other thus confirming the sum of that infinite series is not a definable number!


OK but now he says one method to get the sum of a DIFFERENT infinite series
1 + 1/2 + 1/4 + 1/8 ...
is to get the limit of the average of the sum as you increase n towards infinity where n is the first finite n number of terms of that series.
So for n=2 that is the average of;
1 and 1/2
and for n=3 that is the average of;
1 and 1/2 and 1/4
and for n=4 that is the average of;
1 and 1/2 and 1/4 and 1/8
and so on and you can see that as n increases the average tends to 2 as required. That is fine.
But then he applies the same method to the original infinite series of;
1 - 1 + 1 - 1 + 1 - 1 ...
and notes that using the same method you get the limit of the average of the sum as you increase n towards infinity of 1/2.
BUT the difference here is while no matter how you put an infinite series of brackets around the 1 + 1/2 + 1/4 + 1/8 ... and use another valid method to find its limit you don't get contradictory results, in contrast, as previously shown, you DO get contradictory results if put an infinite series of brackets around the 1 - 1 + 1 - 1 + 1 - 1 ... (with it equaling either 0 or 1 or 1/2).
So I think just because you get a consistent result of 1/2 if you apply this other method (of limit of averages) of finding the limit accounts to nothing because you are still left with the earlier contradictory results that are contrary to it being 1/2 (because they say it is 0 and 1)


The simplest way I can say what I personally think the 'flaw' is of his reasoning is just to simply point out that he ignored the earlier contradiction of that same infinite series both equaling 1 AND zero and then continued regardless even though I think that that proves right from the start that there is no definable number as a limit thus any later argument to define a limit is irrelevant. Do you agree with that?

Well, that is what I make of it.

Originally posted by FabianFnas
When it is about a finite numbers of integers, then the sum is also a integer. So says common sense.
When it is about an in-finite number of integers, then the common sense is all wrong.

Correct. Because of that, you cannot do equations where you equate an infinite series to an integer then expect all to work out well. As pointed out at some of the links, in order to get away with such manipulations, one must subtly redefine certain things in order to hold for infinite series.

It is trivial to prove that the sum of the series 1+2+3+.. cannot possibly be an integer, one need only point out that if it was an integer, then it would be a member of the series and thus smaller than the true sum and thus a contradiction.