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 certain1 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.

$\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!

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.

$\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 $S_3 = \i \frac{\sqrt3}{6}$.

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.

- in fact, probably most [↩]

## TarquinGroup

RT @icecolbeveridge: [FCM] Why the maths of infinite sums is dangerous: http://t.co/Kh4KNYZADZ

## SherriBurroughs

RT @icecolbeveridge: [FCM] Why the maths of infinite sums is dangerous: http://t.co/Kh4KNYZADZ

## BParkEd

RT @icecolbeveridge: [FCM] Why the maths of infinite sums is dangerous: http://t.co/Kh4KNYZADZ

## Robert Anderson

Excellent posts. I’m enjoying them from my phone watching over my 2-day-old baby here. Very clear, and I feel silly for not remembering more from my old Analysis lectures…

## Robert Anderson

And there’s this too http://blog.rongarret.info/2014/01/no-sum-of-all-positive-integers-is-not.html

## Colin

There’s been quite a blogstorm about it! I was just about to add some links.

## Colin

Ah, congratulations! We had a baby boy a month ago – hope you’re getting some sleep 🙂

## 8ctopus

“Why the maths of infinite sums is dangerous – Flying Colours Maths” http://t.co/ezlZage27N

## icecolbeveridge

I’ve rounded up half a dozen posts on the 1+2+3… video and linked to them at the end of this: http://t.co/cOI4X0f7oD . Any I’ve missed?

## icecolbeveridge

@mjb1233 @C_J_Smith I’ve got some comments on that video here: http://t.co/cOI4X0f7oD (TL;DR: result may be ok, method shown is awful).

## mjb1233

@icecolbeveridge @C_J_Smith yeah I agree the method wasn’t exactly rigorous, but they did provide a link to another video…

## mjb1233

@icecolbeveridge @C_J_Smith that does a more rigorous proof (although I haven’t watched it so can’t vouch for it) for those who want it.

## icecolbeveridge

Updated argument/roundup re @numberphile video here: http://t.co/cOI4X0f7oD Let me know if I’ve missed any posts (for or against).

## daveinstpaul

@icecolbeveridge @numberphile This post by Terry Tao from 2010 was very helpful. http://t.co/gTJs37jAFm

## MathsDoctorTV

How dangerous is the maths of infinite sums?http://t.co/QqJ8FGpLbl

## christianp

@icecolbeveridge Ahh, we were going to do one of these for @aperiodical!

## CardColm

@icecolbeveridge @numberphile says “Jordan Ellenberg is a bit mean to Phil.” but is that link there to Jordan?? @JSEllenberg

## JSEllenberg

@CardColm @icecolbeveridge @numberphile Yeah, that link is to @MarkCC — I am going to write about it but haven’t yet!

## evelynjlamb

RT @icecolbeveridge: Updated argument/roundup re @numberphile video here: http://t.co/cOI4X0f7oD Let me know if I’ve missed any posts (for …

## evelynjlamb

@icecolbeveridge There are some physics-based posts from @drskyskull & @blakestacey. I’m on my phone, so I’m not going to try to link them

## oditorium

RT @icecolbeveridge: Updated argument/roundup re @numberphile video here: http://t.co/cOI4X0f7oD Let me know if I’ve missed any posts (for …

## drskyskull

@evelynjlamb @icecolbeveridge Here’s my recent post: http://t.co/V1vavUuMNn & an old one where I do analytic cont: http://t.co/vscCuXn8iq

## drskyskull

@evelynjlamb @icecolbeveridge Here’s @blakestacey: http://t.co/162Iwm6P2R & there are a few others at bottom of my recent post!

## oditorium

.@icecolbeveridge @numberphile great roundup- unfortunately we probably dont even reach -(1+2+3+4+…) times the people who saw the video!

## drskyskull

@evelynjlamb @icecolbeveridge @blakestacey P.S. if you want your head to explode, check out this Physics Buzz piece: http://t.co/FDjXar5zex

Pingback: An infinite series of blog posts which sums to -1/12 | The Aperiodical

## evelynjlamb

It was also called calm by @icecolbeveridge http://t.co/2NzrVcEqpB & someone else I can’t seem to find now.

## MatthewArbo

RT @evelynjlamb: It was also called calm by @icecolbeveridge http://t.co/2NzrVcEqpB & someone else I can’t seem to find now.