Power Series

The central idea

A power series centered at $a$ is $\sum_{n=0}^{\infty}c_n(x-a)^n$. It has a radius $R$ such that it converges absolutely for $|x-a|<R$ and diverges for $|x-a|>R$. Inside that interval it defines a function that may be differentiated and integrated term by term with the same radius.

Definitions, hypotheses, and notation

Uniform convergence on every closed subinterval strictly inside the radius justifies many operations beyond pointwise substitution, including exchanging limits and integrals. Introductory calculations use the termwise rules, but the strict interior condition is why endpoint behavior cannot be carried along automatically.

Algebra of power series resembles polynomial algebra inside common convergence domains. Products use Cauchy convolution of coefficients, and substitutions must map inputs into the base series interval. These domain checks are as important as coefficient arithmetic.

Conceptual meaning

A power series is an infinite-degree polynomial with a precisely controlled domain of convergence. Its coefficients encode local behavior, and finite partial sums provide polynomial approximations whose tails measure error.

A dependable method and decision rule

1. Identify the center by matching powers of $(x-a)$.
2. Use ratio or root analysis to find the radius.
3. Test real endpoints separately for the interval.
4. Differentiate or integrate coefficients and powers term by term only inside the radius.
5. Track constants of integration and redo endpoint tests after transformations.

Fully worked example

Common mistakes and why they fail