Taylor Expansion — How Far Does Increasing the Order Improve the Approximation?

We approximate y = sin x around the origin at order 1, 2, 3, …. As the order increases, you can see the curves overlap even at points far from x = 0. We then look at how far this works, and why there are functions that never get close, however high the order.

Mochizuki Research Group, Tokyo University of Science / Based on the lecture notes “Taylor’s theorem”. The derivation is on the explanation page (in Japanese).