A Taylor series rebuilds a function out of its derivatives at a single point a. Each new term is a higher power of (x − a), so the polynomial hugs the curve outward from the center.
But the hug only reaches so far. Past the radius of convergence R the partial sums stop settling and fly apart — a wall no number of terms can cross.
Cauchy’s surprise: R is the distance from a to the nearest place the function breaks — and that place can hide in the complex plane. 1/(1+x²) is smooth on the whole real line, yet its series dies at |x−a| = √(a²+1), pinned by the invisible poles at ±i.
Slide the center and the wall slides with it. That is the whole secret: the wall tracks the singularity.