Some time ago, I posted a series on the lecture that I’ll give to student to remind them about Taylor series. I won’t repost the whole thing here, but the basic ideas are inductively motivating the concept by starting with a polynomial and then reinforcing the concept with both numerical calculation and comparison of graphs.
After giving this lecture recently, one of my students told me about this terrific video on Taylor series that does much of the same things, with the added bonus of engaging animations. I recommend this highly.
Mean Green Math 