A Follow-Up to The Infinite Series and the Mind-Blowing Result


Phil Plait in Slate:

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, and 3) mea culpa, with a defense (hopefully without being defensive).

More here. Evelyn Lamb offers some criticisms in Scientific American:

Zeno’s paradox says that we’ll never actually get to 1, but from a limit point of view, we can get as close as we want. That is the definition of “sum” that mathematicians usually mean when they talk about infinite series, and it basically agrees with our intuitive definition of the words “sum” and “equal.”

But not every series is convergent in this sense (we call non-convergent series divergent). Some, like 1-1+1-1…, might bounce around between different values as we keep adding more terms, and some, like 1+2+3+4… might get arbitrarily large. It’s pretty clear, then, that using the limit definition of convergence for a series, the sum 1+2+3… does not converge. If I said, “I think the limit of this series is some finite number L,” I could easily figure out how many terms to add to get as far above the number L as I wanted.

There are meaningful ways to associate the number -1/12 to the series 1+2+3…, but I prefer not to call -1/12 the “sum” of the positive integers. One way to tackle the problem is with the idea of analytic continuation in complex analysis.

More here.