Does 1 – 1 + 1 – 1 + 1 . . . . . = 1/2

This morning, for a little first day of school fun, we played with Grandi’s Series.

I’ve seen the series pop up in a few places in the last few days – first in part of a little note I wrote up inspired by a Gary Rubenstein talk:

A Talk I’d live to give to calculus students

and then a day or two later in this tweet from (the twitter account formerly known as) Five Triangles:

Associative property
(1 + -1) + (1 + -1) + (1 + -1) +… = 1 + (-1 + 1) + (-1 + 1) +…
0 + 0 + 0 +… = 1 + 0 + 0 +…
Therefore 0 = 1#mathchat

— 🕔📐 ᴹᴬᵀᴴ𝒞𝒜ℱℰ ☕ ₂.₇₁₈₂₈₁₈₂₈₄₅₉₀₄₅₂₃₅₃₆₀₂₈₇₄₇₁₃₅… (@Five_Triangles) August 28, 2016

So, what’s going on with this series? What would the boys think?

Here’s their initial reaction:

And here’s their reaction when I showed them what happens when we assume that the series does sum to some value x:

We have touched a little bit on this series (and my favorite math term “Algebraic Intimidation”) previously:

Jordan Ellenberg’s “Algebraic Intimidation”

It is fun to hear the boys struggle to try to explain / reconcile the strange ideas in Grandi’s series. I’m also glad that they are learning to think through what’s going on rather than just believing the algebra.