This is a follow-up to last week’s piece on the Numberphile video claiming that $1 + 2 + 3 + 4 + … = -\frac{1}{12}$.

I mentioned something in the last article about certain ((in fact, probably most)) infinite sums not being well-defined, and wanted to add some examples to show how they can be problematic.

Let’s accept - for the sake of this article - that the infinite sums treated in the video hold true:

$S_1 = 1 - 1 + 1 - 1 + … = \frac12$ $S_2 = 1 - 2 + 3 - 4 + … = \frac14$ and $S = 1 + 2 + 3 + 4 + … = - \frac{1}{12}$

I’m going to consider $S_3 = 1 + 1 + 1 + 1 + …$ and show that it’s ‘equal’ to several different values.

Result number 1

$\begin{array}{ccccccc} S & = & 1 & + 2 & +3 & +4 & +…. \\ S & = & & +1 & +2 & +3 & + … \\ \hline S-S = 0 & = & & 1 & + 1 & + 1 & +…&\end{array}$

OK! So $S_3 = 0$. Brilliant. But wait!

Result number 2

Let’s work with $S_1$. Clearly, adding $S_1$ to $S_3$ gives us $S_1 + S_3 = 2 + 0 + 2 + 0 + …. = 2S_3$ – which means $S_3 = S_1 = \frac12$!

It can’t be both, can it?

Oh, but it gets worse.

Result number 3

$\begin{array}{ccccccc} S_3 & = & 1 & +1 & +1 & +1 & +…. \\ S_3 & = & & +1 & +1 & +1 & + …. \\ S_3 & = & & & + 1 & +1 & +…. \\ \hline (1+1+1+1+…)S_3 & = & 1 & + 2 & + 3 & + 4 & +… \end{array}$

Or, if you prefer, $S_3^2 = S = -\frac{1}{12}$. That makes $S_3 = \i \frac{\sqrt3}{6}$.

Result number 4

In fact, you can ‘prove’ that $S=1 + 2 + 3 + 4 + …$ adds up to other things, too. For instance, if you double it, you get $2S = 2 + 4 + 6 + …$. If you add it to itself, but shift the second one along, you get the odd numbers: $2S = 1 + 3 + 5 + 7 + …$. If you add those two together, you get $4S = (1 + 3 + 5 + …) + (2 + 4 + 6 + …) = 1 + 2 + 3 + 4 + … = S$. So, if $4S = S$, $S = 0$ (if it’s defined).

Oh dear. It’s almost as if we shouldn’t be allowed to do regular arithmetic with sums that don’t converge.

Digging a little deeper, Wikipedia makes the strong claim that “stating $1+2+3+4+…=-\frac1{12}$” is an abuse of notation”, which is maths-speak for ‘wrong’. However, there is something in the idea of the sum taking the value of $-\frac1{12}$ in certain contexts.

I’ve not read up on analytic continuation properly (hey! I’ve got a month-old baby to look after) but I gather it’s a method for consistently ‘removing’ infinities to get consistent answers. I have vague recollections of my final year complex analysis course doing something with Laurent series that may be related, but I’m still using the baby excuse to avoid reading up on it.

My problem with the video isn’t that it came up with the answer of $-\frac{1}{12}$. If Ramanujan says it, I’m not going to argue. However, I do have two problems with it:

  • The methods of adding terms and manipulating infinite sums used in the video are at best misleading and at worst irresponsible;
  • Getting rid of infinities is a) a very interesting idea and b) probably not something to be glossed over lightly.

At the very least, the manipulations in the video should come with a health warning.

Some other people have said interesting things:

Terry Tao explains things on a deep level. (Another post from Terry here (thanks, Dave), and a third here.

Stefan (oditorium) explains what’s really happening with the ‘sum’ in a much more accessible way. PhysicsBuzz also has a stab at putting the physics side of the argument.

Ron Garret shows another way that the manipulations don’t work. (Thanks, Robert.) Evelyn Lamb gives probably the calmest explanation of what’s going on.

Evelyn also (thanks!) pointed me at Dr Skyskull has another discussion trying to bridge the analytical point of view with the physical one as well as an an older post on the same topic; and at Blake’s to-the-point takedown.

Richard shows some more examples of why you need to be careful with infinite sums, and has a nice warning label a more ruthless blogger than me would steal.

Phil Plait (Bad Astronomer) discusses the problem… Mark CC (not Jordan Ellenberg - apologies to both, thanks to Colm for pointing it out) is a bit mean to Phil. Phil apologises. Konstantin, meanwhile, is just as cross.

Cathy O’Neill (MathBabe) hits the nail on the head: if it’s hocus pocus, it isn’t maths.

Brady (the man behind the camera defends the piece by saying ‘there’s a more rigorous thing linked’, ‘we’re not a textbook’, and ballsily linking to the wikipedia article saying ‘this is an abuse of notation.’ (My response: analysis is hard enough as it is; probably more people have watched this video than have ever taken an analysis course; you have a responsibility not to spread bad practice.)

* Edited, Jan 18, 21 and 23, 2014, to add links * Corrected misattribution of MarkCC’s post on Jan 23, 2014.