Near a chosen center, a polynomial can match a function’s value and several of its derivatives.
Each added coefficient matches one more derivative at a. Sine and exponential Taylor series converge to their functions for every real x. For log(1+x) centered at a > −1, the series converges to the function only on −1 < x ≤ 2a+1. A finite polynomial is still defined outside that interval, but the series does not converge there.
A Taylor polynomial may be locally excellent and far away very inaccurate.
The full series converges for all real x. Low-degree polynomials can still be inaccurate far from the center.