For polynomials, it's kind of trivial that taylor series work. It's sort of magical in the case of sine and cosine and the exponential function, but that's mor

For polynomials, it's kind of trivial that taylor series work. It's sort of magical in the case of sine and cosine and the exponential function, but that's more a case, i think, of the "magical" properties of those functions than the taylor series. In the general case, taylor series only approximate a function around a point and it's not even guaranteed to do it very well or very far from that point.