Investigating the harmonic series and prime numbers

Fun little milestone today as we hit our 300th Family Math project! The topic for today was the harmonic series and prime numbers – a topic that we’d sort of touched on a few projects ago:

Counting and a Fun Harmonic Series Fact

I began with a quick introduction to the Harmonic series so that the boys would know what we were going to talk through today. The one fact about infinite series that they always seem to remember comes from this old Numberphile Video:

Here’s our quick introductory discussion about the harmonic seris

Next we touched on what it means for a series to add up to \infty. Part of the reason I wanted to talk about adding to infinity was to see if the boys would see that 1 + 2 + 3 + \ldots would also meet my definition of adding up to \infty. They didn’t make that connection, so I’ll have to explore that with them another time.

Also in this section we begin looking into the sum of the harmonic series by dividing the terms into groups whose sum is greater than 1/2.

For the next part of the project we explored some more complicated groups. The ideas here mainly involve estimating fraction sums. There’s also a little bit of introductory counting ideas hanging around, too, I guess.

Next we looked at the sum of the reciprocals of the prime numbers. The surprise here is that this sum is also infinite. I don’t know an easy proof that this sum is infinite. The only proof that I know relies on / introduces generating function ideas and a little bit of Taylor series ideas.

But, knowing this series is infinite leads to some interesting questions, and my older son actually wondered a pretty neat thing about prime numbers that we explore in the final video:

In the last bit of the project we talk a little bit about the number of primes and the distribution of primes. The main topics here are estimating the number of primes less than a certain number, and the question about the number of twin primes.

To have this discussion we have to talk a little bit about logarithms and to simplify that discussion I borrow Jordan Ellenberg’s idea of “flogarithm” which is the number of digits in the decimal representation of an integer.

So, a couple of advanced topics, but a neat conversation. I think that kids can understand some of the basic ideas here and even form some of their own ideas about how to think through these problems. Nice way to celebrate our 300th Family Math project!