Like the Ramones, let’s get straight to the hook:
If you want them to learn infinite series, you’ve got to give them a question. And the way to do that is show them two things that must be connected, but they can’t quite see the connection.
Make them write out the first five terms in an infinite sum in fraction form, so the students see the pattern in the sum. Then you make them generate the terms AND the partial sums in columns of a spreadsheet (much faster than a TI-83). They will recognize some decimals in the sums. That’s when you have them. Why does adding up an infinite number of fractions yield the natural log of 2, or pi over four? It may take a few days to get them there, but now they want to do the journey.
Infinite series & Taylor Series are about as abstract as it gets in high school math, so the spreadsheet quickly adds up hundreds or thousands of terms to really help the students make conjectures. This lesson was developed according to an idea from Peter Mili.