January 20, 2014 | 20

Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter

A Numberphile video posted earlier this month claims that the sum of all the positive integers is -1/12.

I’m usually a fan of the Numberphile crew, who do a great job making mathematics exciting and accessible, but this video disappointed me. There is a meaningful way to associate the number -1/12 to the series 1+2+3+4…, but in my opinion, it is misleading to call it the sum of the series. Furthermore, the way it is presented contributes to a misconception I often come across as a math educator that mathematicians are arbitrarily changing the rules for no apparent reason, and students have no hope of knowing what is and isn’t allowed in a given situation. In a post about this video, physicist Dr. Skyskull says, “a depressingly large portion of the population automatically assumes that mathematics is some nonintuitive, bizarre wizardry that only the super-intelligent can possibly fathom. Showing such a crazy result without qualification only reinforces that view, and in my opinion does a disservice to mathematics.”

Addition is a binary operation. You put in two numbers, and you get out one number. But you can extend it to more numbers. If you have, for example, three numbers you want to add together, you can add any two of them first and then add the third one to the resulting sum. We can keep doing this for any finite number of addends (and the laws of arithmetic say that we will get the same answer no matter what order we add them in), but when we try to add an infinite number of terms together, we have to make a choice about what addition means. The most common way to deal with infinite addition is by using the concept of a limit.

Roughly speaking, we say that the sum of an infinite series is a number *L* if, as we add more and more terms, we get closer and closer to the number *L*. If *L* is finite, we call the series convergent. One example of a convergent series is 1/2+1/4+1/8+1/16…. This series converges to the number 1. It’s pretty easy to see why: after the first term, we’re halfway to 1. After the second term, we’re half of the remaining distance to 1, and so on.

Zeno’s paradox says that we’ll never actually get to 1, but from a limit point of view, we can get as close as we want. That is the definition of “sum” that mathematicians usually mean when they talk about infinite series, and it basically agrees with our intuitive definition of the words “sum” and “equal.”

But not every series is convergent in this sense (we call non-convergent series divergent). Some, like 1-1+1-1…, might bounce around between different values as we keep adding more terms, and some, like 1+2+3+4… might get arbitrarily large. It’s pretty clear, then, that using the limit definition of convergence for a series, the sum 1+2+3… does not converge. If I said, “I think the limit of this series is some finite number *L*,” I could easily figure out how many terms to add to get as far above the number *L* as I wanted.

There are meaningful ways to associate the number -1/12 to the series 1+2+3…, but I prefer not to call -1/12 the “sum” of the positive integers. One way to tackle the problem is with the idea of analytic continuation in complex analysis.

Let’s say you have a function *f(z)* that is defined somewhere in the complex plane. We’ll call the domain where the function is defined *U*. You might figure out a way to construct another function *F(z)* that is defined in a larger region such that *f(z)=F(z)* whenever *z* is in *U*. So the new function *F(z)* agrees with the original function *f(z)* everywhere *f(z)* is defined, and it’s defined at some points outside the domain of *f(z)*. The function *F(z)* is called the analytic continuation of *f(z)*. (“The” is the appropriate article to use because the analytic continuation of a function is unique.)

Analytic continuation is useful because complex functions are often defined as infinite series involving the variable *z*. However, most infinite series only converge for some values of *z*, and it would be nice if we could get functions to be defined in more places. The analytic continuation of a function can define values for a function outside of the area where its infinite series definition converges. We can say 1+2+3…=-1/12 by retrofitting the analytic continuation of a function to its original infinite series definition, a move that should come with a Lucille Bluth-style wink.

The function in question is the Riemann zeta function, which is famous for its deep connections to questions about the distribution of prime numbers. When the real part of *s* is greater than 1, the Riemann zeta function ζ(s) is defined to be Σ^{∞}_{n=1}n^{-s}. (We usually use the letter *z* for the variable in a complex function. In this case, we use *s* in deference to Riemann, who defined the zeta function in an 1859 paper [pdf].) This infinite series doesn’t converge when *s*=-1, but you can see that when we put in *s*=-1, we get 1+2+3…. The Riemann zeta function is the analytic continuation of this function to the whole complex plane minus the point s=1. When *s*=-1, ζ(s)=-1/12. By sticking an equals sign between ζ(-1) and the formal infinite series that defines the function in some other parts of the complex plane, we get the statement that 1+2+3…=-1/12.

Analytic continuation is not the only way to associate the number -1/12 to the series 1+2+3…. For a very good, in-depth explanation of a way that doesn’t require complex analysis—complete with homework exercises—check out Terry Tao’s post on the subject.

The Numberphile video bothered me because they had the opportunity to talk about what it means to assign a value to an infinite series and explain different ways of doing this. If you already know a little bit about the subject, you can watch the video and a longer related video about the topic and catch tidbits of what’s really going on. But the video’s “wow” factor comes from the fact that it makes no sense for a bunch of positive numbers to sum up to a negative number if the audience assumes that “sum” means what they think it means.

If the Numberphiles were more explicit about alternate ways of associating numbers to series, they could have done more than just make people think mathematicians are always changing the rules. At the end of the video, producer Brady Haran asks physicist Tony Padilla whether, if you kept adding integers forever on your calculator and hit the “equal” button at the end, you’d get -1/12. Padilla cheekily says, “You have to go to infinity, Brady!” But the answer should have been “No!” Here, I think they missed an opportunity to clarify that they are using an alternate way of assigning a value to an infinite series that would have made the video much less misleading.

Other people have written good stuff about the math in this video. After an overly credulous Slate blog post about it, Phil Plait wrote a much more levelheaded explanation of the different ways to assign a value to a series. If you’d like to work through the details of the “proof” on your own, John Baez has you covered. Blake Stacey and Dr. Skyskull write about how substituting the number -1/12 for the sum of the positive integers can be useful in physics. Richard Elwes posts an infinite series “health and safety warning” involving my old favorite, the harmonic series. I think that the proliferation of discussion about what this infinite series means is good, even though I wish more of that discussion could have been in the video, which has more than a million views on YouTube so far!

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Luboš Motl and others have deeply, exhaustively demonstrated that the unlimited sum of successive positive integers beginning with 1 is a negative number, -1/12. However, 40 years of contingent exquisite mathematics elegantly quantizing gravitation monstrously fail to afford

empiricalproducts.Everything is consistent, for mathematics is not empirical. If it were, it would be a science.

Link to thisExcellent post, Evelyn. I think you are right.

Link to thisI was confused on the part with 2(S2). The sum varies from 1 to -1 which averages to 0 and divided by 2 is 0. Not 1/2==>1/4

Link to thisGood question. This is one of the problems with the kinds of series manipulations they’re doing. They haven’t told us ahead of time what rules are allowed, so it seems like hocus-pocus even though there’s a way to do it rigorously.

Link to thisThe series you’re asking about is S2=1-2+3-4+5-6 and so on. As you keep summing it up, you get 1, -1, 2, -2, 3, -3, and so on. These average to 1, 0, 2/3, 0, 3/5, 0, 4/7, and so on. The even terms are 0 and the odd terms are approaching 1/2. Depending on what rules you decide are allowed in how you manipulate series, you can decide to take it to the average of these two numbers, which gets you 1/4. You can read more about how this manipulation works (and what the rules are) here: http://en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%C2%B7_%C2%B7_%C2%B7

In general, you get into trouble when you start rearranging the terms of infinite series that don’t converge. It’s very easy to end up with meaningless statements.

Isn’t there an error, because (S2)*(2/2) or [2(S2)]/2 does not return the same result as S2.

Link to thisI’m not quite sure where you’re seeing that. There are issues with these kinds of series manipulations, but I don’t see a place where we have the statement you describe.

Link to thisClearly; the title says…”sum of positive numbers.” They proceed to use negative numbers in their proof. The actual series is negative infinity to positive infinity. Wikipedia says “There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integers {1, 2, 3, …}, while for others the term designates the non-negative integers {0, 1, 2, 3, …}.” So, are they talking about natural numbers or integers? Clearly the set of natural numbers doesn’t include negative numbers; by definition. I always liked math because of its lack of ambiguity. I guess it wouldn’t be entertaining if it was a rigorous proof.Taking artistic license I suppose.

Link to thisThe error in the infinity addition problem is the presenter’s premise that because his first equation alternates between 0 and 1, it is okay for him to average the value to ½ . But that is not always correct. Think about 0 = dead (or even = ‘not pregnant’), and 1 = alive (or = ‘pregnant’). You are either dead or alive, or not pregnant or pregnant, but you are not ½ “dead-alive” or ½ pregnant. Another example is an electric circuit. If the circuit/switch is opened (=0), no electricity flows in the line. However, when the circuit/switch is closed (=1), the light bulb glows. You don’t have ½ open-closed circuit.

Link to thisFor ready reference , I am recapturing the proof as they had proved it

And , I want to show you the various loop holes and deficiencies that lead to such fallacies .

Let me start from the very beginning , where they started from, a very good place to start.

THEY PROVED THAT. [1+2+3+4+5+6 +.....................+ up to infinity ] = [- 1/12 ]

————————————————————————————————————-

They proved it in the following three steps.

Step 1

———-

They started with an infinite series like, S1= 1-1+1-1+1-1+ ……….up to infinity.

And, it was proved by them that S1 = 1/2

What was wrong in their step 1 itself ? (which they continued to repeat in other two steps too)

Many things.

1. they conveniently set aside two more equally valid results,

viz, S1 = 0 and, S1= 1 and , they conveniently selected just one result out of the three,which helped them to prove the result ” predicted “. by them.

The two other results set aside by them were,

(A) that , S1 = 0 ,which would be easily obtained by grouping the terms in pairs

eg. (1-1)+(1-1) + (1-1) + ….up to infinity .

(B) that , S1 = 1, which could be obtained by isolating the first term and grouping the rest in pairs ,

eg. 1- (1-1) -(1-1) -(1-1) – ……… Up to infinity .

(C) they proved S1 =1/2 , by shifting one place to the right or to the left and re- writing the series one below the other and then, adding them up and dividing by 2 .

I will give you here a simpler proof than that for proving S1=1/2 .

It is as follows :

Let S1 = 1-1+1-1+1+ ……to infinity …….now,.isolate the first term,1 and group the rest together

= 1- (1-1+1-1+1-1+…….to infinity )

ie, S1 = 1- S1 ; now, transpose S1 to the LHS to get,

S1 + S1 = 1

2 S1 = 1. Divide by 2 , both sides to get

S1=1/2 ( proved ) .

In this proof , I have isolated one term , grouped the rest and divided by 2

So we found that S1 can have three values

(a) S1= 0

(b) S1 = 1/2.

(c) S1 = 1

2.However ,

The mistake thats being committed in all these procedures ( including in mine too ) were the same which were ,

Isolating a term or

transposing terms ,

shifting a place ,

grouping or

dividing or adding up or multiplying or subtracting an infinite series ,

all of these are NOT PERMITTED in a divergent infinite series because,

(A ) infinity is NOT a Number but is just a profound concept , an idea.

(B) infinity cannot be dissected or cut into pieces as infinity has no component parts .

(C) infinity if added or subtracted from infinity does not become two infinities or zero , but still remains as ONE WHOLE infinity only.

The nature of INFINITY is clearly given in these four lines of the Sanskrit sloka ,

ॐ पूर्णमदः पूर्णमिदम्

पूर्णात् पूर्णमुदच्यते

पूर्णस्य पूर्णमादाय

पूर्णमेवावशिष्यते ||

Meaning,

Om, That which is complete, remains complete,

From the completeness comes the completeness

If completeness is taken away from completeness,

Completeness still remains.

They have repeated the same mistake again in steps 2 and 3 in proving

S2 = 1/4 and then S3= -1/12 .

Lets look at it now.

In step 2 they considered yet another series S2 = 1-2+3-4+5 -6 + ……….to infinity.

And , then proved that S2 = 1/4 corresponding to S1= 1/2 which was one of the three values of S1; conveniently neglecting the other two, viz , S1=0 and S1=1.

If we don’t neglect these two, we get two more values of S2 ,as follows .

Writing S(2) = 1-2+3-4+5 -6 +7……….to infinity……………(a)

Shift a place left & re-write. S(2). = 1-2+3-4+5 -6+ ……….to infinity………(b)

Add (a) +(b), 2S(2) = 1-1+1-1+1-1+1+…..to infinity ,which is our series S1

So we get, 2S2 = S1 which was = 1/2 , or, 1 or 0 as proved in step1 .

Hence , S2. = 1/2 S1

——————-

Therefore S2 = 1/4 if S1 is 1/2 or,

S2 = 1/2 if S1 is 1 or ,

S2. = 0 if S1 is 0 .

( note : there are 3 values for S2 also corresponding to the 3 values of S1 )

Lastly in step 3,we will consider the subject series

S3 = 1+2+3+4+5+6 +……………+ up to infinity

And, will prove that S3 = 0 OR, -1/6 OR , -1/12 .

————————————————————–

Step 3

———

Here is how ,they have proved, S3= 1+2+3+4+5+6 +…………………+ up to infinity = ( -1/12)

————————————————————————————————————————-

S(3) = 1+2+3+4+5+6 +…………………+ up to infinity…………(A)

And , S (2) = 1-2+3-4+5-6 +…………………….up to infinity………….(B)

Subtract (A) – (B)

{ S(3)-S(2)} = 0+4+0+8+0+12+………………….up to infinity

= 4 ( 1+2+3+4+……to infinity )

= 4 S(3). Now,Transpose S1to right and S2 to left

{S (3) – 4 S(3)} = S(2). but S(2) = 1/4 ( proved above @ B )

Therefore , {-3 S3} = 1/4. Divide both sides by 3

Hence , S(3). = – 1/12

Ergo, [1+2+3+4+5+6 +.....................+ up to infinity ] = [- 1/12 ] ………..PROVED .

( note , here also the other 3 values of S(3) have been omitted corresponding to the 3 values of S2 ) .

Summarizing , the values

———————————————-

We proved that S2=1/2 S1 and,S3 = -1/3 S2 or,

S3 = -1/6 S1

Therefore,

If S1=0 , then S2=0 and, S3= 0

If S1 = 1/2 then. S2=1/4 and, S3= -1/12

If S1= 1. then S2= 1/2 and, S3= -1/6

Therefore , S(3) could be = -1/12 OR, -1/6 OR , 0

——————————————————————–

Does it not make -1/12 = -1/6 = 0 ?

(Hope string theory is not based on such absurdities ).

In such cases mathematicians say that the value of a diverging infinite series is indefinite or INDETERMINATE .MEANS, WE CANNOT ASSIGN ANY UNIQUE VALUE TO IT.

no physical result is valid if it’s based on an ” evaluated ” divergent infinite series.

Other similarly absurd results can also be obtained for series S3 =1+2+3+4+5 +….to infinity Group & write 1+2+3+4+5 +….to infinity as = 1+ (1+1)+(1+1+1) +(1+1+1+1) + …….to infinity.

=. 1+ 1 + 1 + 1 +…….to infinity.

And if we re-group the terms and rewrite

S2 =1-2+3-4+5 -6+7 …To infinity as = 1+(3-2 ) + (5-4 ) + (7-6 ) +…..To infinity,

=.1+ 1 + 1 + 1 + ……to infinity…..(S4)

And,if we regroup & rewrite S2 = 1 -2. +3. -4. +5 – To infinity

As = 1+(- 1-1)+(1+1+1)+(-1-1-1-1)+(1+1+1+1+1)+..to infinity

And regroup again as = 1 + (-1-1+1) +( 1+1-1-1-1)+(-1) + (Next five +s and six

negatives to get -1),and so on ……..To infinity

= 1 -1. -1. -1. -1 ….to infinity… (S5)

Therefore ,(S4) = (S5) = (S2) = (S3)

ie, 1+1+1+1 +1 +…To infinity = 1-1-1-1-………………..to infinity

= 1+2+3+4+5 +……….to infinity,

= 1-2+3-4+5 -6+7 ….. To infinity

All of which become = 0, or, 1/2 or,1/4 or, -1/6 or, -1/12

Doesn’t it make the number line itself vanish or restrict itself to a negative length ?

Which is probably exactly what happens at infinity !

We can prove that S2 = S1 too.

———————————————————–

Start with S2 as. = 1- 2. + 3. -4. +5 -6. +7 ….. To infinity, Rewrite S2 & group as= 1-1-1. + 1. +1+1 -1. – 1-1-1 +1+1. +1+1+1 -1-1-1-1-1-1 … To infinity.

This will ,by proper grouping , reduce to , ( first term 1, next three to get -1 ,next three to get +1 , next five to get -1 and so on , to get the series S1 which was equal to .

S1 = 1. -1. +. 1 -1 . +1. …….To infinity

= 0 OR , 1/2 OR , 1 ( we proved this earlier ) .

Hence, we have, S1 is = S2 is = S3

Each of which is. = 0 or, 1/2 or, 1 or,1/4 or,-1/6 or ,-1/12 ! ( a set of six values )

Which would mean that -1/12 =-1/6= 0= 1/4= 1/2 =1 ( absurd and incompatible with our number system , but might be compatible with infinity ) .

Hence is it rightly said by MATHEMATICIAN , ABEL that , I quote ,

“The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes …”unquote .

( by the way, wolfram alpha doesn’t even attempt to evaluate the sum of divergent infinite termed series and rightly too , as absurd results can only be arrived at , which cannot be applied to any physical situation ) .

Having said that , it’s interesting to consider what could these bizarre results actually mean , if at all it ever means anything?

After all it’s mathematics, an abstract human thought form and must have validity in some domain, if not in the physical domain.

Some physicists in their over-enthusiasm to breath some ” physical” meaning (or nearly analogues to physical meaning), to their mathematical models , say that it’s all true in some dimensions that transcend our physical 4 dimensional space- time . They propose “many worlds”, “alternate universes” , “multiverses” all of which lie suspended for ever beyond our sensual perceptions somewhere in the femto-cosm,or below Planck dimensions .

Probably their cumulative effect on our senses is to cut our universe to size , enabling us to perceive ,by our macro- physical senses, as a “4-D space- time cross- section” of a multidimensional Cosmos with some reasonable degree of consistency on which we base our physical laws and manipulate material world for our survival and entertainment .

No wonder that under psychotropic drugs our view of even our normally solid physical universe become totally disoriented and distorted as perhaps mind takes over matter then !

Then it transcends 4D space-time universe into dimensions of awareness that we in our physical form cannot reach .

Drug addicts find such alternate universes comfortable than our rigid physical universe .

Beware of infinity ; treat it with respect , its NOT a number and, any digit in a

Link to thisdivergent infinite series does NOT acquire its digital positional value on the number line .

On the other hand , It then will acquire a different “place value” which is actually infinity !

For one twelfth of a dollar I will buy the world from a mathematician.

Link to thisvamsi krishna thummala

your answer is wrong

1-2+3-4+5-6…………………………………………

1-(1+3)/2 +3 -(3+5)/2 +5 -(5+7)/2……………………………….

1-1/2-3/2+3-3/2-5/2+5-5/2……………………………………..

=1/2

(1+2+3+4+………………………..=s)-(1-2+3-4+5-6…………………=1/2)

2(2+4+6+………..)=s-1/2

4(1+2+3+………….)=s-1/2

4s=s-1/2

s=-1/6

1+2+3+4+…………..=-1/6

Link to thisgreat article … I’d seen this argument in a physics course, and it was used in the derivation of the number of dimensions the universe must have according to some variant of string theory, and I found (and still find) it a bit precarious. But at least analytic continuation is a well-defined mathematical concept, while the approach shown in the video is not.

Any of us, using basic logic, could prove that you cannot make a number negative by adding more positive terms to it … and so, as you rightly say, seeing this argument just makes the derivation seem like very arbitrary use of whatever interpretation is convenient.

What would be useful would be to start with the analytic continuation argument, and then show the infinite series bit as a way to “rationalise” how infinite series can produce unexpected results …

Link to thiswhat is the value of 1^2+2^2+3^2+…….?

Link to thisi think it is 1/28

Here is a proof by contradication that 1+2+3+4+… is not -1/12.

1+2+3+4+… = 1+3+5+7+… + 2+4+6+8+…

1+3+5+7+… = 1+1+1+1+… + 0+2+4+6+…

1+3+5+7+… = 1+1+1+1+… + 2*(1+2+3+4+…)

so, 1+2+3+4+… = 1+1+1+1+… + 4*(1+2+3+4+…)

1+1+1+1+… = -3*(1+2+3+4+…) = 3/12

But also, 1+2+3+4+… + 1+1+1+1+… = 2+3+4+5+…

so 1 + 1+2+3+4+… + 1+1+1+1+… = 1+2+3+4+…

1 + (-1/12) + 3/12 = -1/12

1 + 3/12 = 0

nope

Link to thisSo, to take issue with the S1 part of the proof:

s1=1-1+1-1+1…

Isn’t this the same as saying infinity times one plus infinity times minus one? Which seems to me to equal exactly zero in all cases. Infinity minus infinity.

I’m not as mathematically adept as most of you, so if I’m wrong, please let me know how. I love to learn.

Link to thisInfinity isn’t a number that we can manipulate exactly the way we manipulate finite numbers. Indeed for finite numbers, 6-6=0 and a million minus a million is 0. But infinity just doesn’t work like that. If you took precalculus or calculus and dealt with limits, you were actually doing this kind of work, figuring out what expressions that involve infinity or division by 0 mean.

For example, say you have 6x/x, and you want to know what happens as x gets bigger and bigger (“goes to infinity”). As x gets bigger, the top and the bottom both grow without bound, so you’re dividing infinity by infinity. If you divide a regular number by itself, you always get 1 (6/6=1), but in this case, we can see that for any finite x, the expression 6x/x, reduces to just 6. The x’s cancel out. So here, the top grows 6 times as fast as the bottom, so this instance of “infinity/infinity” has a limit of 6. But if you substituted a different number in for 6, you could make “infinity/infinity” equal to whatever number you wanted. Mathematicians therefore call infinity/infinity an indeterminate form. Infinity minus infinity is another indeterminate form (and there’s a way to transform an expression that looks like infinity-infinity to one that looks like infinity/infinity), so its limit could be any number at all, or the limit could not exist.

It’s also important to note that when it comes to infinite sums, we can’t always rearrange them the way we would a finite number of terms. If you have three numbers, it doesn’t matter what order you add them up in, but in the case of infinite sums, doing that can change the answer. So grouping all the positive numbers and all the negative numbers together isn’t a “legal” move. If you want to see that in action, look up conditionally convergent series or the Riemann series theorem.

In the case of S1, the traditional and more intuitive way of assigning a limit to an infinite sum doesn’t yield an answer, so deciding that it equals 1/2 is an alternative way of assigning a value to the expression. It’s valid, but you need to specify the way you did it. (In this case, we can get the value 1/2 by using techniques called Cesaro summation or Ramanujan summation, if you’d like to look those up.)

Link to thisthis can also be applied to renormalization and integrals

http://en.wikipedia.org/wiki/Renormalization#Zeta_function_regularization

to extend the defintion to divergent integrals INt(0,oo) x^{m}dx

Link to thisNo animals (or intellects) were harmed in the making and streaming of those Numberphiles episodes. It is not often that a math story has that “wow!” factor, and if something attracts attention to math with a memorable zing, that has to be a good thing. Or so it seems to me.

Link to thisUsing these and similar methods, it is not difficult to see that 0 = 1, a concise result, worth committing to memory, which can be handy in a variety of apparently unrelated proofs. It is just a nice little identity to keep in your back pocket. I plan to rely on in the (unlikely) event that I freeze up while defending my thesis.

Link to thisYou’ve put this issue into perspective, and it was an enjoyable read. Saying that the sum is simply -1/12 was misleading, and I’m glad that you clarified that. Having an alternative definition of infinite sum made me want to unify the two definitions somehow. Following your links I came across an article by Terence Tao, and saw the asymptotic expansion of divergent series. I saw that it had an unbounded term and a finite term. The unbounded term of course swamps out the finite term, but choosing only the finite term appears to be equivalent to redefining the type of summation. This reminds me somewhat of choosing the real or imaginary part of a complex number, or of the fourth root of unity including +/-i, once imaginary numbers were discovered/invented. I wondered what you would think of that.

Link to this