how do i use a taylor series centered at some x value to approximate the value of the function centered at a different x value?

Question

for example, if im given some taylor series centered at 5 of f(x) but i want to find f(3), how do i do this? (btw that example is not my homework problem, i'm just trying to understand the method)

thanks alot!

Answer 1

It depends on the radius of convergence. If the series has a radius of convergence of ∞ such as the series for sin(x), cos(x), and exp(x), you can evaluate f(x) where x equals any defined value, even though the series are expanded about an arbitrary value x0. In the three examples cited above, they are expanded around 0.

If the radius of convergence is small, and the desired evaluation point is not within the radius of convergence, then you'll need to expand the series about an appropriate point. But then you'll also need to calculate f'(x0), f"(x0), .... but that's part of the game of series expansion.