An infinite series of blog posts which sums to -1/12

Many of you who are aware of the internet will have noticed that some mild controversy has surrounded a recent Numberphile video, posted last week:

Briefly, the video claims the counter-intuitive result that that $1+2+3+4+\ldots=-\frac1{12}$. The value of $-\frac{1}{12}$ can be obtained using a method called analytic continuation, whose purpose is in effect to allow one to assign values to functions at points where they are otherwise undefined. This is a clever and interesting branch of maths, but given its aims, it is perhaps less surprising that it manages to assign a value to an otherwise undefined sum. However, the video only very faintly alludes to this concept, instead ‘proving’ the equation using purely arithmetic techniques. The analogues of these techniques over in analytic-continuation-land may be valid, but the tricks themselves, as they are shown in the video, are not. A twenty-minute accompanying video spells out a few of the details, undoing some of the damage (for the small percentage of people who go on to watch it).

The internet’s response to the video was swift and merciless. In particular, some criticism was directed at a post on Slate, which linked to the video and described the content, in a way which some people argued didn’t capture the nuance of how the video was WRONG. The Slate piece, written by Phil Plait, has been followed by a second piece which retracts some of the statements made in the first article. There have also been apparently several million other blog posts on the topic.

In an effort to reassure you, our readers, that we here at the Aperiodical noticed this extravaganza of mathematical mud-slinging, here’s a round-up of some of our favourite blog posts discussing it. Think of it as being like the Carnival of Mathematics, only it’s the Carnival of $1+2+3\ldots$ “=” $-\frac{1}{12}$.

Phil Plait’s Slate article, introducing the video:
Infinite series: When the sum of all positive integers is a small negative fraction.

Responses

A response to Phil Plait’s article:
On -1/12, adding infinitely many numbers, and Phil Plait’s rash and incorrect claims | konstantinkakaes.com

Phil Plait’s response to the responses to his article:
Follow-up: The Infinite Series and the Mind-Blowing Result.

A response to the video from Aperiodifriend Colin Beveridge:
Why I don’t buy that $1 + 2 + 3 + … = -\frac{1}{12}$ – Flying Colours Maths | Flying Colours Maths

A follow-up in which Colin variously breaks maths:
Why the maths of infinite sums is dangerous – Flying Colours Maths | Flying Colours Maths

A response to the video by Cathy O’Neill at Mathbabe:
If it’s hocus pocus then it’s not math | mathbabe

A response to the video, with a nice analogy:
“AMAZING” EQUALITY DEBUNKED | Colin Grove

A follow-up from Colin Grove, with more detail:
AMAZING EQUALITY UPDATE | Colin Grove

A response to the video and related responses, which takes no prisoners:

A response to the video, from Dr Skyskull: Infinite series: not quite as weird as some would say | Skulls in the Stars

Evelyn Lamb’s remarkably calm response to the video:
Does 1+2+3… Really Equal -1/12? | Roots of Unity, Scientific American Blog Network

Richard Elwes reiterates an illustrative example in response to the video:
Richard Elwes – Infinite Series – A Health and Safety Warning

Ron Garrat’s response to the video:
Rondam Ramblings: No, the sum of all the positive integers is not -1/12

Ron Garrat’s response to the responses to his response to the video:
Rondam Ramblings: Does it matter if the sum of all integers is -1/12?

A well-explained, if somewhat harsh, response to the video:
oditorium U – The non-sense of claiming 1+2+3+…=-1/12

Terry Tao’s G+ post in response to the responses:
Terence Tao – Google+ – There is a lot of discussion in various online mathematical…

Brady’s response to all the responses:
Periodic Videos: Thanks for the messages

Tony Padilla (the guy in the video)’s response to everything:
What do we get if we sum all the natural numbers?

Older posts on the topic

An old post by Dr. Skyskull:
Infinite series are weird — redux! | Skulls in the Stars

An old post by Terry Tao on the topic:
The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation | What’s new