# Sum of Infinite Series

I'm not sure which category this post actually belongs to, or if the title of this post is even accurate. I guessed Calculus was the closest one.

I watched this video on the web after a professor told me this mathematical phenomenon (http://www.youtube.com/watch?v=w-I6XTVZXww‎). It asserts that 1+2+3+4+5+6+7+8+9+... equals -1/12. In fact, this is the basis of String Theory, and it is agreed upon by mathematicians. Intuitively, I don't understand how adding positive numbers to infinity can equal a negative number.

Can someone simply explain this (or at least conceptualize this) to a grade 10 student? Thanks.

P.S. Here's a Wikipedia article: http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

That video has become very famous by now. And I think it is said because it is misleading and does a great disservice to mathematics.

First of all, of course the sum of all positive integers is not a negative number. If anything, the following sum is true

$$1+2+3+4+5+6+... = +\infty$$

which is exactly what we expect.

Let me explain this entire situation with another example. Maybe you already know of geometric series, but the theory says that

$$1+\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... = 2$$

This makes sense. If I take 2 empty glasses, and if I fill the first glass with water, then the second glass half with water. Then I fill in the half of the remaing emptiness, and so on. In the limit (which is not something we can do in real life) you will have filled up 2 glasses.

Similarly, we can show that for any ##x\in (-1,1)## holds that

$$1+x+x^2 + x^3 + x^4 + ... = \frac{1}{1-x}$$

What about other ##x##? Well, the above equality is not true for them, and besides, the proof only works for ##x\in (-1,1)##.

OK, but what if I pretend like the above equality still holds? Then for example, you would get things like

$$1+2+4+8+...= -1$$

which of course isn't true, since the above equality only holds for ##x\in (-1,1)##. But still, these kind of results tend to be useful and handy. So whenever you read

$$1+2+4+8+ ... = -1$$

then you shouldn't be reading this as an equality. Rather, you should read it as

$$1+2+4+8+... = -1$$

under the conditions that the above equality is true for all x. So it's not a true equality, but a pretend-equality. Mathematicians have all kind of fancy names for those, for example, they say that

$$1+2+4+8+ ... = -1$$

in the 2-adic number system (don't worry if you don't know what that is). So again, it means the equality usually is false, but we consider some other kind of "equality" where it is true.

Same thing with

$$1+2+3+4+5+... = -1/12$$

This is a false equality under the usual definition of equality. But we say that the equality is true with Ramanujan summation. This means that we change our definition of equality a bit so that this becomes true. This new definition of equality might or might not have anything to do with reality or intuition, that is the important thing to realize!

Apparently, this kind of "summation methods" are useful in string theory, but I can't comment on that.

Curious3141
Homework Helper
I predict this thread is going to recur ##\displaystyle \aleph_0## times, just like the "##\displaystyle 0.999\dots = 1##??" threads. I predict this thread is going to recur ##\displaystyle \aleph_0## times. I predict that I will probably make an FAQ on the topic instead...

The fact that mathematicians abuse the fact that their "equality" is in a different number system to make cool Youtube videos seems rather... sleazy.

I predict this thread is going to recur ##\displaystyle \aleph_0## times, just like the "##\displaystyle 0.999\dots = 1##??" threads. Honestly, this topic is more worthy for discussion than an equality that college freshman don't like because it "looks weird."

lurflurf
Homework Helper
= means different things at different times, that is in no way sleazy or a great disservice. We might say the meaning of = should be given each time, but that is very burdensome and context suffices.

examples
7 apples=7 oranges
= same number of fruit

7 apples!=7 oranges
= amount of orange juice produced by smashing

2001=1
= mod 2

2001!=1
= integers

sqrt(2)/2=1/sqrt(2)
= real numbers

sqrt(2)/2!=1/sqrt(2)
= same score on elementary algebra exam

There is no usual =. Different = are needed for different situations.

= means different things at different times, that is in no way sleazy or a great disservice.
Haha, of course you are correct. But what I meant is that the people of the video never said what the meaning of = is. They didn't even mention that = could have different meaning to begin with, like you explained so well in this post. That is what I find sleazy and a great disservice! And yes, context suffices. But this video is watched by students in tenth grade, so I guess it would have been better to be a bit more explicit about the context in the video!

= means different things at different times, that is in no way sleazy or a great disservice. We might say the meaning of = should be given each time, but that is very burdensome and context suffices.

examples
7 apples=7 oranges
= same number of fruit

7 apples!=7 oranges
= amount of orange juice produced by smashing

2001=1
= mod 2

2001!=1
= integers

sqrt(2)/2=1/sqrt(2)
= real numbers

sqrt(2)/2!=1/sqrt(2)
= same score on elementary algebra exam

There is no usual =. Different = are needed for different situations.
Abusing the ambiguity when context is not provided (or worse, when context is provided, and is misleading) in order to make 'cool YouTube videos' is sleazy. I don't know how that translated into "it is sleazy for = to mean different things at different times."

The mathematician actually has the audacity to say "no" when the camera man suspects that 1+2+3+... tends to infinity. Why is that incorrect to say? Oh right, because that different (and connotatively the one that it seems like we are actually talking about) use of "=" does not make for a cool and surprising YouTube video that will get shares on Facebook. Sleazy.

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Ok, thank you for your replies. Does this mean that 1+2+3+4+5+...=−1/12 is actually just a mathematical hoax, that the youtuber had no legitimate intention in his "educational" video, even though this equation (or whatever it's called) is used in many physics textbooks and theories (e.g. string theory)?

Ok, thank you for your replies. Does this mean that 1+2+3+4+5+...=−1/12 is actually just a mathematical hoax, that the youtuber had no legitimate intention in his "educational" video, even though this equation (or whatever it's called) is used in many physics textbooks and theories (e.g. string theory)?
No, it means that while the -1/12 result does have a meaning, it is not that this series literally converges to that value.

The fact that mathematicians abuse the fact that their "equality" is in a different number system to make cool Youtube videos seems rather... sleazy.
The mathematician actually has the audacity to say "no" when the camera man suspects that 1+2+3+... tends to infinity. Why is that incorrect to say? Oh right, because that different (and connotatively the one that it seems like we are actually talking about) use of "=" does not make for a cool and surprising YouTube video that will get shares on Facebook. Sleazy.
I just want to point out that both of the men in the video are physicists. Unfortunately for mathematicians (and mathematics) a great deal of "popular" mathematics on the internet is being "taught" by non-mathematicians; these gents, Vi Hart, and Salman Khan come to mind. They more often than not sacrifice rigor for the sake of entertainment or ease of understanding. They occasionally stray into the realm of lying about math or even outright making **** up and trying to pass it off as "real" math to their unsuspecting audience. And then when folks call them on their B.S., they turn into common bullies, accusing the dissenters of having their "knickers in a twist" and other such nonsense.

It's infuriating.

Ok, thank you for your replies. Does this mean that 1+2+3+4+5+...=−1/12 is actually just a mathematical hoax, that the youtuber had no legitimate intention in his "educational" video, even though this equation (or whatever it's called) is used in many physics textbooks and theories (e.g. string theory)?
No, the equation is very true, but it should be interpreted correctly. Summation methods such as zeta regularization (that is used here) are very important in mathematics, but it must be dealt with with a lot of care. So again, under the "usual" rules, we have

$$1+2+3+4+5+6+... = +\infty$$

But if we change the rules a little bit, then we have

$$1+2+3+4+5+6+... = -1/12$$

Mathematicians love changing the rules a little bit and see what they get. This is perfectly ok, as long as it is spelt out what the new rules are. In this case, the new rules are probably a bit complicated to explain to a high-school student (sadly), but they are all consistent.

So mathematicians are perfectly allowed to play by new rules if they explain rigorously what they are. However, these new rules might or might not be intuitive or useful. In this case, it is faaaar from intuitive, but it tends to be quite useful.

It is certainly not a hoax. It is something that is false under the usual, common sense rules. But it is true under a new set of rules.

Maybe a comparison. Usually, something like ##24=0## is completely and utterly false. It's nonsense. But what if we change our rules a bit. So let's do that. We do that by defining equality as ##x=y## if and only if ##24## divides ##y-x##. We will denote this by ##x\cong y## (to be clear in this case. Some people do use = here).

So in this case ##24\cong 0##. Indeed, ##24## divides ##24-0##.
Same thing: ##26\cong 2##. Indeed, ##24## divides ##26-2##.

We get some very weird number system here (called "integers modulo ##24##"). But it is entirely consistent. So you get weird stuff like ##12+12 = 0## and ##4\cdot 3 = 0##.

This is certainly nonintuitive. But it is useful! I claim you use this every day of your life. When? Whenever you read the clock. Indeed, you are very well aware that when the clock strikes ##24## hours, then it is actually ##0## hours.
And you know that if it is ##4## (am), and you drive in your care for ##30## hours, then it will be
$$30 + 4 \cong 10$$
hours. So this is again a useful system that looks false at the first sight. The same thing is true with

$$1+2+3+4+5+6+... = -1/12$$

So if you can make up some consistent rules and if you play with them, then you can get all sorts of exotic mathematics that might actually be useful!

• 1 person
No, the equation is very true, but it should be interpreted correctly. Summation methods such as zeta regularization (that is used here) are very important in mathematics, but it must be dealt with with a lot of care. So again, under the "usual" rules, we have

$$1+2+3+4+5+6+... = +\infty$$

But if we change the rules a little bit, then we have

$$1+2+3+4+5+6+... = -1/12$$

Mathematicians love changing the rules a little bit and see what they get. This is perfectly ok, as long as it is spelt out what the new rules are. In this case, the new rules are probably a bit complicated to explain to a high-school student (sadly), but they are all consistent.

So mathematicians are perfectly allowed to play by new rules if they explain rigorously what they are. However, these new rules might or might not be intuitive or useful. In this case, it is faaaar from intuitive, but it tends to be quite useful.

It is certainly not a hoax. It is something that is false under the usual, common sense rules. But it is true under a new set of rules.

Maybe a comparison. Usually, something like ##24=0## is completely and utterly false. It's nonsense. But what if we change our rules a bit. So let's do that. We do that by defining equality as ##x=y## if and only if ##24## divides ##y-x##. We will denote this by ##x\cong y## (to be clear in this case. Some people do use = here).

So in this case ##24\cong 0##. Indeed, ##24## divides ##24-0##.
Same thing: ##26\cong 2##. Indeed, ##24## divides ##26-2##.

We get some very weird number system here (called "integers modulo ##24##"). But it is entirely consistent. So you get weird stuff like ##12+12 = 0## and ##4\cdot 3 = 0##.

This is certainly nonintuitive. But it is useful! I claim you use this every day of your life. When? Whenever you read the clock. Indeed, you are very well aware that when the clock strikes ##24## hours, then it is actually ##0## hours.
And you know that if it is ##4## (am), and you drive in your care for ##30## hours, then it will be
$$30 + 4 \cong 10$$
hours. So this is again a useful system that looks false at the first sight. The same thing is true with

$$1+2+3+4+5+6+... = -1/12$$

So if you can make up some consistent rules and if you play with them, then you can get all sorts of exotic mathematics that might actually be useful!
I don't know who you think you are, but here in America, our clocks use mod 12. • 1 person
Could someone explain why this result is actually useful in string theory, I've always wondered.

lurflurf
Homework Helper
Abusing the ambiguity when context is not provided (or worse, when context is provided, and is misleading) in order to make 'cool YouTube videos' is sleazy. I don't know how that translated into "it is sleazy for = to mean different things at different times."
The problem seems to be writing
$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$$
$$\zeta(s)=\text{[analytic continuation of]}\sum_{n=1}^\infty \frac{1}{n^s}$$

Though if = means "has the same analytic continuation as" the two are the same.
The mathematician actually has the audacity to say "no" when the camera man suspects that 1+2+3+... tends to infinity. Why is that incorrect to say? Oh right, because that different (and connotatively the one that it seems like we are actually talking about) use of "=" does not make for a cool and surprising YouTube video that will get shares on Facebook. Sleazy.
If we are picking nits throwing around infinities is at least as bad. I would agree the video could have explained things more clearly. I don't think saying "analytic continuation or zeta regularization" three or four times would have effected the views that much, most viewers would probably not even notice. Physics has a different approach and different standards. The video was very informal.

The rules by which they bend maths are not explained, are not obvious to, atleast, me and the claim of the sum 1+2+3+4.. =/= ∞ make for enough for me to not appreciate the video or share it for that matter. Were they so consistent in playing by their own rules, perhaps they should have let the audience in on it too.

Another video from the same guys that I believe clears it up:

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The problem seems to be writing
$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$$
$$\zeta(s)=\text{[analytic continuation of]}\sum_{n=1}^\infty \frac{1}{n^s}$$

Though if = means "has the same analytic continuation as" the two are the same.
I am not sure what you're saying here. It seems like you are advocating that we do not properly define our terms.

Yes, obviously if the = sign means exactly what it has to mean to make them correct then they are correct in any circumstance. They never mentioned, not once, what their notation meant. Therefore it is only reasonable to assume that their notation was to be interpreted in the standard way (the sum converges to this value.)

If we are not clear in our mathematics then it is meaningless. We are all aware that = is just a symbol and can mean different things it to in a different situation, but we need to define it.

If I said 2 = 44 would you evaluate that as a false statement? I didn't even indicate what I meant by =, therefore instead of writing down the infinite relations we could impose on = for making it true, as the writer, the fault is on me, and I simply wrote a false, nonsense statement because you have no choice but to assume I meant numerical equality for the statement to have any information in the first place.

If we are picking nits throwing around infinities is at least as bad.
You're kidding, right?

The rules by which they bend maths are not explained, are not obvious to, atleast, me and the claim of the sum 1+2+3+4.. =/= ∞ make for enough for me to not appreciate the video or share it for that matter. Were they so consistent in playing by their own rules, perhaps they should have let the audience in on it too.
Precisely.

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lurflurf
Homework Helper
I am not sure what you're saying here. It seems like you are advocating that we do not properly define our terms.
Definitions are good. It would have been better in my opinion to mention them. Definitions are a bit like an onion. If we define zeta summation we should define analytic continuation. Then we should define analytic. We should define convergent series and divergent series. Maybe define sum and integers. Think of the purpose and audience. I quote the mathematical criminal creator from http://www.nottingham.ac.uk/~ppzap4/response.html .
" I wanted a simple presentation that people with limited mathematical training could follow, so I took the plunge and I went with it." and "There is an enduring debate about how far we should deviate from the rigorous academic approach in order to engage the wider public. From what I can tell, our video has engaged huge numbers of people, with and without mathematical backgrounds, and got them debating divergent sums in internet forums and in the office. That cannot be a bad thing and I'm sure the simplicity of the presentation contributed enormously to that. In fact, if I may return to the original question, "what do we get if we sum the natural numbers?", I think another answer might be the following: we get people talking about Mathematics. "

If we are not clear in our mathematics then it is meaningless. We are all aware that = is just a symbol and can mean different things it to in a different situation, but we need to define it.
Being clear is an admirable ambition. It is also possible get lost in details. You or I would have made the video differently. Perhaps the creators would have done some things differently if they redid it. I think it was great that they got a lot of people talking about interesting mathematics even if they had to gloss over some technical points. A treatise on divergent series full of definitions and proofs is unlikely to go viral. Perhaps a handful of viewers will be inspired to read such a treatise now or explore mathematics in some other way. Even books on mathematical topics omit details. Hopefully there is a footnote and references. The videos has notes and references, some were added later, but they are there now.
If I said 2 = 44 would you evaluate that as a false statement? I didn't even indicate what I meant by =, therefore instead of writing down the infinite relations we could impose on = for making it true, as the writer, the fault is on me, and I simply wrote a false, nonsense statement because you have no choice but to assume I meant numerical equality for the statement to have any information in the first place.
That is not what they did. They clearly demonstrated what they meant. If you showed the way in which 2=44 I would be satisfied.
lurflurf said:
If we are picking nits throwing around infinities is at least as bad.
You're kidding, right?
Of course not, how can I know what infinity means when you did not define it? Even if you defined your infinity it is not a very helpful answer. I again quote Dr. Tony Padilla
"The answer we gave was, to the surprise of many I'm sure, −1/12. It's by no means obvious, but this is the only sensible value one can attach to this divergent sum. Infinity is not a sensible value." Perhaps "sum of the divergent series" is unfortunate terminology. Maybe if we called it the signature, shadow, residual, or something else it would help.

Definitions are good. It would have been better in my opinion to mention them. Definitions are a bit like an onion. If we define zeta summation we should define analytic continuation. Then we should define analytic. We should define convergent series and divergent series. Maybe define sum and integers. Think of the purpose and audience. I quote the mathematical criminal creator from http://www.nottingham.ac.uk/~ppzap4/response.html.
Not at all, you don't need to be overly precise here. You could just say that there is "some sensible way of extending ##\zeta##". You don't have to go into what that way is. Being vague is ok in my book, as long as you indicate when you're being vague!

Being clear is an admirable ambition. It is also possible get lost in details. You or I would have made the video differently. Perhaps the creators would have done some things differently if they redid it. I think it was great that they got a lot of people talking about interesting mathematics even if they had to gloss over some technical points. A treatise on divergent series full of definitions and proofs is unlikely to go viral. Perhaps a handful of viewers will be inspired to read such a treatise now or explore mathematics in some other way. Even books on mathematical topics omit details. Hopefully there is a footnote and references. The videos has notes and references, some were added later, but they are there now.
Right. But there is certainly a way here that you can be clear and not get lost in the details. The video was certainly not clear.

That is not what they did. They clearly demonstrated what they meant. If you showed the way in which 2=44 I would be satisfied.
Sure, you would be satisfied. But you also have an extensive mathematical background! The people who watch the video do not have such a background. You seem to be forgetting this.

Of course not, how can I know what infinity means when you did not define it? Even if you defined your infinity it is not a very helpful answer. I again quote Dr. Tony Padilla
"The answer we gave was, to the surprise of many I'm sure, −1/12. It's by no means obvious, but this is the only sensible value one can attach to this divergent sum. Infinity is not a sensible value." Perhaps "sum of the divergent series" is unfortunate terminology. Maybe if we called it the signature, shadow, residual, or something else it would help.
Infinity is a perfectly sensible answer to this question. Even if you don't define infinity, most people will find

$$1+2+3+4+5+... = +\infty$$

to be intuitively obvious. I don't get why you would even need to define it. You're putting the cart before the horse here, and I'll tell you why. The results like $$2^n\rightarrow +\infty$$ and

$$1+2+3+4+5+... = +\infty$$

should come first and only then should we attempt to define infinity. Our definition of infinity will then be tested by applying it on the previous results. So we defined infinity exactly in order to obtain the previous results. If our definition of infinity did not allow us to obtain these results, then the definition would be useless and would be changed rather quickly by a more useful one.

You see, the way mathematics grows is that we first encounter formulas like

$$1+2+3+4+5+... = +\infty$$

and only then we attempt to define the relevant terms such as ##+\infty## and infinite sums. Likewise, calculus was developed far before a rigorous definition of the real numbers. And the real numbers were developed exactly in order to obtain the results of calculus! We would never be interested in some kind of definition that would not give us the intuitive results.

So no, it would be silly to first need to define ##+\infty## in order to explain to the public that

$$1+2+3+4+5+... = +\infty$$

We can of course do so, but I doubt anybody would care (why should they anyway)...

Anyway, the OP has not been back, so I'm locking this thread.