A power series has a region where its infinite sum defines a function.
The finite sum satisfies (1−x)S_N=1−x^{N+1} when x≠1. The remainder tends to zero exactly for |x|<1. At x=1 the sums grow, and at x=−1 they alternate between 1 and 0. Endpoints must be checked separately in general power-series problems.
A finite polynomial can be evaluated outside the convergence interval even though the infinite series fails there.