I got a nice question from an old high school classmate asking about yesterday’s blog post on prime numbers and infinity. He asked me how you prove that the sum of the inverse squares was pi^2 / 6 and that the sum of the inverse primes was infinity.

Unfortunately both proofs are a little too advanced for kids. The first requires understanding of the Taylor series for sin(x) (at least the proof that I know), and the second requires a bit of number theory. I wasn’t trying to present the ideas as something for the kids to prove, but rather just trying to show some fun facts that will come down the road (way down the road in this case).

However, there is one series that is pretty easy to understand that also has an infinite sum -> 1 + 1/2 + 1/3 + 1/4 + 1/5 + . . . .

After the question yesterday I toyed around with the idea of doing a second video, but decided to do it another time. Then, as with yesterday’s blog, some funny coincidences happened today. The first being that I started a new section on fractions with my younger son and the second one being a short proof of the divergence of the above sum that Dave Radcliffe put on twitter:

If S = 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + … then
S > 1/2 + 1/2 + 1/4 + 1/4 + 1/6 + 1/6 + …
hence S > S, contradiction.

I’d never seen this clever little trick before and couldn’t wait to get home to show it to the boys:

I’ve been following a bunch of math folks on Twitter for about a year now and just can’t believe how many fun examples I’ve found to share with the boys. This little community on twitter has been a great resource for me.