Follow-up: The Infinite Series and the Mind-Blowing Result

Phil Plait in Slate:

Infiniteseries_question.jpg.CROP.original-originalYesterday, I posted an article about a math video that showed how you can sum up an infinite series of numbers to get a result of, weirdly enough, -1/12.

A lot of stuff happened after I posted it. Some people were blown away by it, and others… not so much. A handful of mathematicians were less than happy with what I wrote, and even more were less than happy with the video. I got a few emails, a lot of tweets, and some very interesting conversations out of it.

I decided to write a follow-up post because I try to correct errors when I make them, and shine more light on a problem if it needs it. There are multiple pathways to take here (which is ironic because that’s actually part of the problem with the math). Therefore this post is part 1) update, 2) correction, 3) and mea culpa, with a defense (hopefully without being defensive).

Let me take a moment to explain right away. No, there is too much. Let me sum up*:

1) The infinite series in the video (1 + 2 + 3 + 4 + 5 …) can in fact be tackled using a rigorous mathematical method, and can in fact be assigned a value of -1/12! This method is quite real, and very useful. And yes, the weirdness of it is brain melting.

2) The method used in the video to write out some series and manipulate them algebraically is actually not a great way to figure this problem out. It uses a trick that’s against the rules, so strictly speaking it doesn’t work. It’s a nice demo to show some fun things, but its utility is questionable at best.

3) I had my suspicions about the method used in the video, but suppressed them. That was a mistake.

More here.