Introduction to infinite series

We’ve been studying series and sequences for a few weeks (with about a 1 week interruption for Thanksgiving) and we finally had time to do a short project tonight.

I started off tonight’s project by asking my son to talk about the Harmonic series and give two different “proofs” for why it diverges:

Next I pulled one of the exercises out of his book. This series is a slightly difficult integral – and we’ve skipped the techniques of integration section for now – so this part was a nice integration review, too:

Following the approach via the integral test, I wanted to show how the comparison test works. The comparison test is actually the topic for tomorrow, so this was new material. He did a nice job thinking through how the test would work:

Finally, I picked a random problem from the book and asked him to work through it. It turned out to be a pretty neat problem asking why the integral test wouldn’t work on a particular series:

It is fun going through sequences and series with my son. It is going to take a bit longer than I initially thought, so we’ll probably be working through the ideas here through mid December. We’ll come back to techniques of integration after that.