Power Series Definition

Calculus 2 · Axiom Academy

LESSON Power Series Definition The general form, the center, the radius of convergence, and the interval of convergence — built from one concrete series. A power series centered at a is built by stacking terms, each a coefficient c_n times a power of the distance (x-a) from the center. The animation builds it one term at a time — notice every term is anchored to the same center. A finite polynomial is defined everywhere, but a power series only converges for certain x . Take a concrete one centered at a=2 : Watch its partial sums as N grows. Where they settle onto a finite value, the series converges ; where they run off to infinity, it diverges . Every power series falls into exactly one of three cases for how big that convergence set can be. The animation grows each set out from its center a — a single point, a finite band, or the whole line. Converges only when x=a . Here R=0 . Example: . Converges when for some . Example: , R=3 . Converges for all x . Here . Example: . The radius of convergence R is the distance from the center a within which convergence is guaranteed. For our series, the Ratio Test makes it concrete: the ratio of consecutive terms is The animation slides a test point x outward from a=2 and tracks that ratio live. The series converges while the ratio is below 1 , and that threshold is hit exactly at distance R=3 .

This is the written version of the interactive lesson above. See the full Calculus 2 course.