"Astounding: 1 + 2 + 3 + 4 + 5 + ... = -1/12"

[Eviscero]

but that's my point, you can't prove that there is no cap on the amount of things there can be.

What do you mean by "there can be?" Of course you get into trouble when you make ambiguous statements.

Do you mean there is a cap on the "amount of things" that can exist in the universe? <- unknowable.

Do you mean there is a cap on the "amount of things" that can be conceived of? <- that would be false.

Proof:
Assume there is a cap on the amount of things that can be conceived of. Say that cap is M. Then the number M+1 cannot be part of those M things which can be conceived of since certainly the whole numbers 1 through M are conceivable and there are M of them. But I can conceive of the number M+1, thus our assumption was false.

[Ekto]

Fun video and fun YT channel. If you are a student in math or are interested by paradoxes and false proofs, you should check out the other videos in the user's channel. Anyway, on to explaining this particular video... (FYI, some calculus or analysis is probably needed to understand the details, but anyone with some mathematical maturity should be able to follow the gist.) I have bolded some important statements that summarize the post.

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False proofs like this one are interesting exercises for beginner students since spotting the flaw is sometimes tricky. Particularly popular ones on Internet message boards are usually disguises of division by zero.

The false proof in this video is more advanced since you have to know about infinity, limits, infinite sums, and the properties thereof. The actual answer is that S=1+2+3+... is +infinity. You should have understood that the answer can certainly not be -1/12. (How you can add strictly positive numbers and get a negative number?)

So how does the video supposedly prove that S=-1/12? Recall some terminology from the video, since I will be using it.

S = 1+2+3+4+...
S1 = 1-1+1-1+1-1+...
S2 = 1-2+3-4+5-6+7+...

The first thing to note is that S1 and S2 are both divergent series (they don't even converge to +inf or -inf). The usual arithmetic operations of addition, multiplication, etc. are *not valid* for divergent series. (For those who have done calculus II, in general they are valid only for absolutely convergent series.) So it is invalid to add S1 to S2 or to subtract S2 from S.

Don't get me wrong. Obviously, the video is meant to make you question where the mistake is. These men, if they really are who they say they are, obviously know that S=+inf, and they know what the mistake is.

So where is the mistake exactly? What is the precise step in which the entire proof goes wrong? It is stating that S1=1-1+1-1+1-1+...=1/2. Remember, infinite sums are *defined* as the limit of the partial sums. So what are the partial sums of this series? As the narrator says, the partial sum of an odd number of terms is 1 and the partial sum of an even number of terms is 0. So the sequence of partial sums is {1,0,1,0,1,0,...}. The infinite sum is defined as the limit of this sequence. But the sequence {1,0,1,0,1,0...} has no limit (the limit superior is 1 and the limit inferior is 0; they are unequal). So we say that S1 does not exist, it's as simple as that.

Now, even if S1 did exist, the rest of the false proof is still not valid because we cannot add and subtract series that do not converge absolutely. (We also cannot rearrange the terms of a conditionally convergent sum without changing the sum itself!) For example, the series M=1-1/2+1/3-1/4+... does converge, and it converges to M=ln(2). If we rearrange the terms to N=1+1/3-1/2+1/5+1/7-1/4+...(so we add two terms with odd denominators, then subtract one term with an even denominator), then you can actually show that N=(3/2)ln(2)=(3/2)M. It is actually different from M, even though the two series have the same terms. Why? Remember that infinite sums are defined as limits of partial sums. The partial sums of these two series have different limits. (Note that M and N are not absolutely convergent since the series 1+1/2+1/3+1/4+... is +infinity. That is a crucial property since rearrangements of absolutely convergent series all converge to the same sum.)

Why does the narrator say that S1=1/2? Well, he says we should just average 0 and 1 to get 1/2. This procedure is actually called Cesaro summation, and it has different properties from regular summation. Regular summation attempts to compute the sum A=a1+a2+a3+a4+... by the limit of partial sums, whose sequence is {s1, s2, s3, s4,...}. Regular summation computes the limit of the sequence s(n) as n->inf and says that is the sum A. Cesaro summation adds a step. It first forms the average of partial sums. So we look at the sequence {s1, (s1+s2)/2, (s1+s2+s3)/3, ...}. The limit of this new sequence is then said to be the Cesaro sum.

You can show that if a series is regularly summable it is also Cesaro summable and the two sums are equal. However, there are series that are *not* summable but nonetheless Cesaro summable. The series S1=1-1+1-1+1-1+... is one such series, and, indeed, its Cesaro sum is 1/2. The rest of the video is still wrong since we cannot add and subtract and rearrange the terms of divergent series. They are invalid operations. I suppose if you wanted to somehow redeem the video, you should work only with Cesaro sums. However, you can show that S=1+2+3+... has a Cesaro sum of +infinity and the alternating series S2=1-2+3-4+5-... is not Cesaro summable. Maybe someone can come up with another false proof to show that S=-1/12 using Cesaro sums?
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So what is the moral? When dealing with infinity, be careful. The arithmetic of things involving infinite limits is emphatically not the same as the arithmetic of finite quantities and finitely many terms. Infinite sums and the like are defined precisely in terms of limits and sequences, and you must work with those definitions to get correct results.

[Arkenaw]

All this proves to me is that either these guys have no idea what they're doing, or everything we think we know about math is wrong.

[Eviscero]

Since a sum is commutative, I'm allowed to switch the order of numbers,

This is the heart of the matter. The order matters for (some) infinite sums. The distinction is made by two notions: conditionally convergent vs. absolutely convergent series.

A remarkable fact is that if an infinite sum converges conditionally one can rearrange the terms of the sum to converge to ANY number they choose. Whereas if a sum converges absolutely, then all rearrangements converge to the same thing! i.e. commutativity holds for absolutely convergent infinite sums.

[heef]

Infinity doesn't exist in real life.

Sure it does. It's just a concept. It is not, however, an element of the set of real numbers, and when people try to treat it as such, dumb frequently happens.

[niil945]

False premise results in false conclusions, astounding!

[Socialhealer]

as soon as you turn a number or answer into a letter such as abc xyz, the most common ones, or in this case s, thats where the bullshit can begin, because you start going oh yeh s2 -s = s, because if you did that with the numbers it wouldn't work.

but to be honest that math just sounds like religion, god does/does not exist and you cannot prove otherwise, then use infinity which we cannot put a value to therefore doesn't exist but in theory does exist, and all you end up with is bullshit really.

[Ekto]

Holy wow this is such a silly thing to do. Step 1 and they have already made a very strange choice. They define the infinite sum

S1 = 1-1+1-1+1 ... to be 1/2 because it is the AVERAGE of the two partial sums that can possibly be achieved: 0 and 1. Okay, fine... you're welcome to do that. I much prefer the standard approach which is to say an infinite sum is said to converge if the limit of the sequence of partial sums converges (that is define the sequence s(k) = the sum of the first k terms. If for every number epsilon > 0 there exists a natural number N such that |s(N) - L| < epsilon, we say the infinite sum "is equal" -- really converges -- to the number L). If no such L exists we say the sequence diverges (in other words is not equal to anything). There are some very interesting results about when an infinite sum converges and it should be pretty clear that S1 does not converge according to the definition proposed above. There are many reasons why the above definition "is the right one" and I am very curious to know in what context their averaging definition "is the right one."

(I bolded the specific part I am replying to.)

What they do is called Cesaro summation. (Look up the Wikipedia article. I can't post it since I have only posted a few times on MMO.)

If a series is summable, it is also Cesaro summable, and the two sums are equal. The converse is not true. That is, a series can be Cesaro summable but not summable. The alternating series S1=1-1+1-1+1-1+... is one such series, and its Cesaro sum is 1/2. The whole idea of Cesaro summation is to extend the notion of summability. It does enjoy many of the same properties as regular summation, while also making certain divergent series now Cesaro-convergent.

One of the most common (maybe most important) applications of Cesaro summation is in the convergence of Fourier series. Consider a periodic function F on the interval (-pi, pi). (I assume you know some analysis from some of your posts. I'm not sure.) The Fourier series of F does not necessarily converge pointwise to F. We need some conditions on the smoothness of F. One such criterion is the Dini criterion, which requires that F have integrable difference quotients. The Cesaro sum of the Fourier series of F, however, requires fewer smoothness assumptions and, if it does Cesaro-converge, it converges to F. (That's pretty amazing.)

The partial summation of a Fourier series is usually effected by integration against the Dirichlet kernel, which has many undesirable properties. (In particular, its L1 norm tends to infinity.) The partial Cesaro-summation of a Fourier series is effected by integration against the so-called Fejer kernel, which has many desirable properties. (In particular, it is a so-called "approximation to the identity".)

You should look up the relevant Wikipedia articles for more information. Fourier series and their properties are typically covered in an intermediate real analysis course if you want to learn more. (I don't know your background.) The course also typically covers topics like Lebesgue measure and Lp spaces.

[Piglord]

What the hell is this?

[Ekto]

All of math is "religious" in a sense. It begins with an existing, fundamental set of conditions that we agree are true without any proof and the rest are absolute consequences of those conditions.

But if you decide to change those conditions, you can have an entirely "new" math. The difference from religion is that both the "original" and the "new" math may have applications.

That's the issue with this video. It's making a claim about a result from a "new" math and trying to tell it's viewers that it's true in "original" math.

This is partially true. The axioms we assume at the foundational level must also be consistent. (That is, it is impossible from these axioms to prove some statement P and its negation ~P (not P).) Unfortunately, proving consistency is often very difficult and actually cannot be done within the same logical system unless that logical system is inconsistent. (This is one of Godel's incompleteness theorems. The exact statement and what we mean by proving from within a system are both much more complex than I let on.)

(FYI, the most common axiom system (Zermelo-Frankel set theory with axiom of choice) has not been proven to be consistent. In fact, it has been proven that, assuming the other axioms are consistent, adding the axiom of choice does not make the system inconsistent. That is, the axiom of choice is consistent with the other axioms, and the negation of the axiom of choice is also consistent with them. (Not at the same time, obviously.) It has also been proven that the axiom of choice is not a logical consequence of the other axioms either.)

Anyway, if you change the axioms, you don't necessarily get a "new" math. You can end up getting an inconsistent set of axioms, from which all statements may be logically deduced. That's not really math. (Also, technically, unless you then change one of the inconsistent axioms, changing the other axioms doesn't give you anything new. All inconsistent logical systems are pretty much the same.)

The last part of the quoted post above is spot on. The video is a false proof and the authors intend it to be confusing and puzzling. They want students of math or anyone interested to figure out and understand what goes wrong. The problem is that pinpointing the mistake requires some knowledge you would not learn until at least Calculus II. So for the layman, the video is a good springboard to discussions about infinity. Unfortunately, most of that discussion on message boards tends to be uninformed and/or incendiary. =(

[Karrion]

http://www.youtube.com/watch?v=E-d9mgo8FGk

Another video by them on the same subject. I think it is a bit more informative than the first one.

[Ekto]

I'm being concise because I'm not responding to mathematicians, but laymen. It is unfortunately vague, but it avoids my post from being skipped due to math-phobias.

Yeah, I understand and got that from your post. I can be here though to expand on shorter posts for those who don't have mathphobia and dare read on.

[Locksrus]

Math itself is just a concept, not a truth

[Grimbold21]

A thread about math.

Goodbye.

[darxide]

"Look at this video from these professional mathematicians who have devoted their entire lives to math. I don't like their answer so I'm going to tell you why these brilliant people are wrong." - The Internet

This is the intellectual equivalent of:

[Socialhealer]

"Look at this video from these professional mathematicians who have devoted their entire lives to math. I don't like their answer so I'm going to tell you why these brilliant people are wrong." - The Internet

well their maths is useless what if i offer to purchase, everything in the universe, and more...an infinite purchase if you will since we don't know where the universe ends just like their sum, it'll cost -1/12, so you actually owe me(this is not serious before you lose your shit over it btw)

stupid maths is stupid, adding positive numbers forever = a negative, sorry but I've already filled my reading bullshit capacity for the week.

[Herecius]

Man, there are a lot of backseat mathematicians and quantum physicists on this forum.

[Caliph]

I assume some of you have seen this already...


Now I'm not a math wiz, but is this not inherently flawed in assuming that 1+1-1+1-1...=1/2? That is just a way for the measurer to make infinity finite in order to work with it, is it not? Thus ruining the whole problem.

this is why i don't do math. It's so stupid. If it's infiinity how can it equal -1/12. And why is he adding 2s1 and then subtracting it from s2? why isn't he adding the numbers straight across to infinity? And how can you add all the way to infinity? so stupid. Math is bullshit, nothing more nothing less. just bullshit.

[Mormolyce]

Now I'm not a math wiz, but is this not inherently flawed in assuming that 1+1-1+1-1...=1/2? That is just a way for the measurer to make infinity finite in order to work with it, is it not? Thus ruining the whole problem.

I can't watch that video here at work but yes that is false because that series does not converge to anything. Specifically:

SUM(i=1 -> infinity) [ (-1)^(i+1) ]

Doesn't converge.

Similarly 1+2+3+4+... = SUM(i=1 -> infinity) [ i ] which clearly diverges.


Want to see something cool that's not fake?

1=0.99999...

Proof:

1/3 = 0.33333...

multiply both sides by 3.

1=0.99999...

QED.

Not a trick, it is 100% true.

[schwarzkopf]

OSo to everyone thinking that the infinite sum of the natural numbers equals -1/2, you are in the wrong and this video is badly explained.

There is an 'extras' video showing that this is just the simplest to explain proof, and that there are other more rigorous proofs available that require a greater understanding of maths.

Suggest watching the extra bit.

Also - maths doesn't have to match common sense, it isn't as if our brains evolved to understand maths at this level.