Polynomial approximations and radius of convergence
A Taylor series approximates a smooth function by an infinite polynomial. The radius of convergence R determines where the series converges:
For entire functions (sin, cos, exp), R=∞. For ln(1+x), R=1 — the singularity at x=−1 limits convergence. For arctan(x), R=1 due to poles at ±i in the complex plane.
Notice the Gibbs-like overshoot near the boundary of convergence.