By Don Koks, 2016.

(or, does 1 + 2 + 3 + ... equal −1/12?)

No, of course they can't. 1 + 2 + 3 + ... has no sum, or we might say that it sums to infinity. The
real question is: why do some people write 1 + 2 + 3 + ... = −1/12? The answer to that involves some
maths, some physics, and some analysis of common misunderstandings about what mathematicians and physicists are
saying. Some physicists mistakenly believe that mathematicians have summed the series to give
−1/12. And some mathematicians mistakenly believe that physicists have summed the series experimentally
to give −1/12. Neither are right, but so much finger pointing of each to the other discipline has
occurred that many laymen now believe that maths *and* physics have both proved that the sum is
−1/12. The subject is an old one, but gained a new lease of life in 2014 with the appearance of a
notorious youtube clip presented by an academic well outside his zone of expertise, who succeeded in proving only
that breaking the rules of elementary mathematics in the age of the Internet can still get you 15 minutes of fame.

Of course, it's physically impossible to use a calculator, abacus, or pen and paper to actually sum a series
that is infinitely long, so mathematicians long ago realised the expression needed to be carefully defined if it
were to have any useful meaning. They define it in a way that matches everyone's expectation of what such an
expression *should* mean: begin the addition term by term in the order written, and keep an eye on the
running sum (also known as the "sequence of partial sums") as each term is added. If this running sum gets
ever closer to some number, then that number will be unique and is called the sum of the series. If the
running sum doesn't behave in that way, then we say the series has no sum. If you start with 1, then add 2
(running sum is 3), then add 3 (running sum is 6), then add 4 (running sum is 10), those partial sums 1, 3, 6, 10,
get bigger and bigger and don't get arbitrarily close to any number at all. So the series 1 + 2 + 3 + ... has
no sum. But you knew that anyway.

But didn't the mathematicians Euler and Ramanujan sum the series to give −1/12? Ramanujan's letter of almost a century ago to the mathematician Hardy, in which he wrote the sum, dates from a different time. Euler's interest was similar to that of Ramanujan: he wanted to see where the rules of mathematics could take him, so he assumed that the sum existed and performed some mathematical gymnastics to arrive at −1/12. Euler and Ramanujan certainly had their feet on the ground enough to know that putting one orange into a big pit, followed by 2 more oranges, then 3 more oranges, and so on forever, is not going to result in there being −1/12 oranges in the pit. They were trailblazers of other times, and they went very far by experimenting with the fewer boundaries that existed back then. Since then, proper boundaries have been drawn, and modern mathematics knows perfectly well where these lie: those boundaries were established by setting axiomatic properties of numbers that keeps mathematics from running off the rails and all hell breaking loose. Euler's early work belongs to his time and is part of mathematical history. He was allowed to do what he did, but modern mathematicians and physicists no longer work under the paradigm that was current in Euler's time. They now work under established rules that weren't available to Euler.

The reason why some modern physicists think that mathematics *has* summed the natural numbers actually
has nothing to do with simple algebraic manipulations of the series. So let's take it in stages, and begin
with some much simpler ideas.

For example, what do we mean by the repeating decimal 0.3333....? This represents the infinite series 3/10
+ 3/100 + 3/1000 + .... It can be shown that the ordered sequence 0.3, 0.33, 0.333, 0.3333, ... converges to
1/3 (or "has a limit of 1/3", or "tends toward 1/3"), and so we define 0.3333... to *equal* 1/3.
(Technically, that means we can always find an element in that sequence of 0.3, 0.33, 0.333, 0.3333, ... which is
as close to 1/3 as we wish, and such that all successive elements lie even closer to 1/3.) By the same token the
repeating decimal 0.9999... equals 1, because the ordered sequence 0.9, 0.99, 0.999, 0.9999, ... converges to
1. In contrast, the ordered sequence 1, 1 + 2, 1 + 2 + 3, ... has no limit at all that you can find on a
number line. By convention we say that it tends to infinity; although you can't find infinity on a number
line, mathematicians do supplement the number system with a "number" called infinity, and so it can be said that 1
+ 2 + 3 + ... equals infinity.

How about the series *1 + x + x ^{2} + x^{3} + ...* where x is a real number? This
series only converges when

If we *were* to assume that 1 + 5 + 5^{2} + 5^{3} + ... equals some number that we can
point to on the real-number line, then it would be easy to find that number. Call it *S*:

S = 1 + 5 + 5Now write^{2}+ 5^{3}+ ... .

5S = 5 + 5and subtract the first line from the second to give^{2}+ 5^{3}+ 5^{4}+ ... ,

4S = −1and conclude that

This idea is routinely analysed using partial sums, and then the reason for why it doesn't work becomes very
obvious. First denote the partial sums by S_{n}, which is certainly possible because they are just
normal everyday sums, each of which makes perfect sense:

SThen_{n}= 1 + 5 + 5^{2}+ 5^{3}+ ... + 5^{n}.

5Sso that subtracting the first line from the second gives_{n}= 5 + 5^{2}+ 5^{3}+ 5^{4}+ ... 5^{n+1},

4Sand_{n}= 5^{n+1}− 1 ,

Now let's picture a scenario, one that more usually occurs in the realm of complex numbers, but we can also explore it in the realm of real numbers. The example here is simple enough to highlight the main points, but don't be misled by its simplicity into believing that more complicated examples are as transparent as this one.

I have a function *f(x) = 1 + x + x ^{2} + x^{3} + ...* that I have worked out is defined for

But as I dabble with various expressions, I happen to notice that the expression *1/(1 − x)* equals
the sum of my series *f(x)* for all *|x| < 1*. I also know that *1/(1 − x)* is
defined for all other values of *x* except *x = 1*. It turns out that of all "sufficiently
smooth" functions, *1/(1 − x)* is the unique function with these properties. For suppose, on the
contrary, that I was able to find another function *g(x)* that also matched my series *f(x)* for
all *|x| < 1*. That would mean *g(x) − 1/(1 − x)* was zero everywhere
inside *|x| < 1*. But if *g(x) − 1/(1 − x)* is "sufficiently smooth", it turns out
that it will have to equal zero *everywhere*. But that means *g(x) = 1/(1 − x)* and so I
haven't really found another function—I've only found *1/(1 − x)* again. So *1/(1
− x)* is a unique extension of my function *f(x)* to the wider world of all *x*
(except *x = 1*). This *1/(1 − x)* is called the *analytic continuation*
of *f(x)*. We have started with a series *f(x)* valid in a limited interval *|x| < 1*,
and managed to come up with a unique function that agrees with *f(x)* in that interval, but is also valid on
a bigger interval.

The crucial point here is that *1/(1 − x)* is *not* the sum of *1 + x + x ^{2} +
x^{3} + ...* outside the interval

f(x) = 1 + x + xI can then correctly write^{2}+ x^{3}+ ... for |x| < 1, f(x) = 1/(1 − x) for all other x except x = 1.

The same ideas apply more generally to functions defined over complex numbers. A "sufficiently smooth"
function is called *analytic*. The series *f(z) = 1 + z + z ^{2} + z^{3} + ...*
certainly exists for all complex numbers

This series *1 + z + z ^{2} + z^{3} + ...* was a simple example. Long ago
mathematicians became interested in the following series:

1 + 1/2This series converges only when the real part of^{z}+ 1/3^{z}+ ...

ζ(z) = 1 + 1/2This series doesn't converge when real(^{z}+ 1/3^{z}+ ... provided real(z) > 1.

It turns out that the zeta function can first be analytically continued to the region real(*z*) > 0
with *z* not equal to 1, with the following series:

1 ζ(z) = ----- (1 − 1/2A second analytic continuation now extends the definition of the zeta function to all^{z}+ 1/3^{z}− 1/4^{z}+ ... ) provided real(z) > 0 and z =/= 1. 1−2^{1−z}

ζ(z) = 2(2π)where^{z−1}sin(πz/2) Π(−z) ζ(1−z) , provided real(z) < 1,

It turns out that when *ζ(z)* is analytically continued in the way of the above,
that *ζ(−1) = −1/12*. Of course, this has nothing to do with the original series that
started *ζ(z)* off, which was only defined when the real part of *z* was greater than 1.
But picture a mathematicians' party where the joke goes around that if we set *z = −1* in the
series *1 + 1/2 ^{z} + 1/3^{z} + ...*, and then say that the result
equals

Of course, mathematicians are well within their remit to search for any connection that might exist between two
functions when one is the analytic continuation of the other. That's a very interesting technical question,
and one that currently has no answer. In other words, how is the most general definition
of *ζ(z)* related to the restricted definition *1 + 1/2 ^{z} + 1/3^{z} + ...*
that holds only for real(

In the meantime, I don't think anyone ever sets *z* to 0 in the series *1 + 1/2 ^{z} +
1/3^{z} + ...* and equates the answer to

Other ideas of summing infinite series exist. One that you will find mentioned in connection with summing
the natural numbers is "Cesaro summation", in which we write a sequence of arithmetic means of the partial sums of
the infinite series. If that sequence converges to some limit, then that limit is called the Cesaro sum of
the infinite series. Cesaro sums can sometimes be better behaved than the usual type. Also, any series
that sums in the usual sense of partial sums will be Cesaro summable to the same number. But some series that
are not summable in the usual sense *are* Cesaro summable—although the series 1 + 2 + 3 + ... is not
one of those. In principle there is an infinite number of different "flavours" of Cesaro sum that one could
define for any one series, purely because there is an infinite number of different ways to define an average: the
arithmetic, geometric, and harmonic means are only three examples of an infinite number of different means. Cesaro
summing uses the usual arithmetic mean, but there is no reason why we shouldn't use any other flavour of mean
instead.

Here's an example of a series that is not summable in the usual sense, but *is* Cesaro summable:

1 − 1 + 1 − 1 + 1 − 1 + ...The sequence of running sums is 1, 0, 1, 0, 1, 0, ... and these average to 1/2, so the Cesaro sum is 1/2 even though the usual sum doesn't exist. Does the Cesaro sum have any physical use here? Note that if you give a man 1 dollar in the first second, then 1/2 dollar in the next second, then 1/4 dollar, then 1/8 dollar and so on, he will eventually be able to spend a sum of money arbitrarily close to $2 (but not more than that) if he waits long enough. But if you give him 1 dollar, then take it away, then give it back, then take it away, repeating these actions indefinitely, he'll never be able to spend anything at all, unless he spends $1 very quickly and then sells what he bought for $1 so that he can give it back to you. He certainly won't be able to spend 1/2 a dollar and keep what he buys. So whereas the usual way of summing says the series has no sum—which matches something about the real world here—the Cesaro sum does not match anything about the real world.

Cesaro summing is really one version of a more general idea in which we define a function, say *f*,
that takes the elements of the infinite series in order and returns a number that we find interesting. Those
who wish to say that the sum of the natural numbers is *ζ(−1)* are really defining such a
function via "*f(1,2,3,...) = ζ(−1)*". That can certainly be done, but is it useful? There
are any number of infinite series in *z* that we could write down that reduce to 1 + 2 + 3 + ... when
some *z* is set to some number, and the result won't necessarily be −1/12. So now we'll have to
introduce *another* function, call it *g*, such that *g(1,2,3,...)* equals whatever number is
returned by that new series of interest. An infinite number of functions will need to be invented to cover
all series. The subject of infinite summation will then be running off the rails, and departing from the very
basic idea that a sum is all about piling objects on top of each other and seeing what, if anything, is taking
shape.

It's sometimes argued that if the series 1 + 2 + 3 + ... has no sum in the usual sense, that we are free to
define it to equal *something* of use. The analogy is used that the square root of 2 doesn't exist in
the rational numbers, so we extend those numbers to become the real numbers, in which case the square root of
2 *does* then exist. Likewise the square root of −1 doesn't exist in the real numbers, so we
extend the real numbers to become the complex numbers, and then the square root of −1 does exist. So,
some argue, we should be allowed to *define* 1 + 2 + 3 + ... to be −1/12. But this line of
reasoning has no logical content. The only way that the square root of 2 and the square root of −1 were
given meaning was precisely by extending the realm of known numbers to a larger set that contained new
entities. For example, we can't just say the square root of 2 equals 4/5; we have to invent a new type of
number, an irrational number. Likewise we can't just say the square root of −1 equals −5 or
349. We need to extend the realm of numbers in order to give it meaning, which we do by inventing a new
number, *i*, for the square root of −1. Likewise, we cannot set 1 + 2 + 3 + ... equal to a real
number, because we already know that 1 + 2 + 3 + ... doesn't converge to any such number. So can we extend the
complex numbers by defining a new object that equals 1 + 2 + 3 + ...? Certainly, and this was done long ago: that
new object is "infinity". Defining it hasn't proved as fruitful as defining irrational numbers and complex
numbers, but it has been done. But infinity does not equal −1/12. Maintaining that an infinite
sum is just too inconvenient and must be replaced by some real number is about as daft as demanding that Earth's
radius should be redefined to be 10 metres to make it quicker to travel to another country.

Some mathematicians and physicists think that physicists have measured 1 + 2 + 3 + ... to equal −1/12 in the laboratory. But no such measurement has ever been made; nor will it ever be made. This measurement is described in the FAQ entry What is the Casimir Effect?. It concerns a prediction made by the Dutch physicist Hendrick Casimir in 1948. Casimir analysed two parallel plates using the language of quantum mechanics. He considered a field that might fill the universe, and in particular the space between the plates. This field was purely a theoretical construct until Casimir came along.

The usual line of reasoning says that the value of this field in some region is to be interpreted in a quantum mechanical sense as being related to the probability of finding a related particle in that region. Since presumably no particles can exist at the plates themselves, the field is required to vanish at the plates. If the field is constrained that way and Fourier-analysed into a sum of "modes", then only a countably infinite number of these modes can exist between the plates.

Casimir calculated the quantum mechanical energy of the mode-restricted field between the plates. As expected, it was infinite, because quantum mechanics allocates a unit of energy to each mode, and there are an infinite number of modes between the plates. His calculation arrived at the sum "1 + 2 + 3 + ...", which expressed this infinite energy.

The idea of infinite energy is something of a problem in practice. What Casimir did next got him through
that problematic infinity. First, he made the series somewhat like a geometrical one by inserting a "cutoff"
factor that could go to 1 in an appropriate limit. This new series was well defined and able to be summed
exactly to some number *S _{L}*, where

This difference in energies was treated like a difference in potential energies. Now, potential energy is
only defined up to an additive constant, so there is no unique potential energy for, say, a mass in a gravitational
field (unless we stipulate the constant, which we do in practice). But the *difference* between two
potential energies certainly is well defined and related to forces. So the difference in field energies with and
without the plates predicted the existence of a tiny force between the plates, and the existence of this force is
known as the Casimir Effect. In fact, a force is a *spatial* gradient of potential energy, but spatial
gradients actually have nothing to do with the difference in field energies with and without plates, so this very
standard explanation of the Casimir Effect doesn't quite work. But it has become standard in the field.

Historically, that's where the Casimir Effect stands, but the above procedure has been re-interpreted by many practitioners in the following way. They note that the original divergent sum (i.e. with plates present) will give what I called the Energy Difference if they simply replace its 1 + 2 + 3 + ... part with −1/12. I think that's of no great consequence: the mathematics of the sums in the Casimir Effect is not complicated, so it should not be surprising to find that the Energy Difference has a lot in common with the non-divergent overall factor in the original divergent sum.

But notice what this interpretation has done: it has taken a divergent quantity and observed that replacing its
1 + 2 + 3 + ... part by −1/12 gives the difference between two related sums (*S _{L}*
and