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

Graphical or geometric meaning

Within the convergence interval, partial-sum polynomials follow the function more closely as degree grows. Outside the radius, individual terms fail to produce a convergent total even if a short polynomial looks reasonable.

Common mistakes and why they fail

Verification and reasonableness checks

• Differentiate a derived series back to a known base series.
• Substitute $x=a$ to verify the constant coefficient.
• Test a point inside and outside the proposed radius.

Manipulate terms only inside the radius

A power series $\sum a_n(x-c)^n$ behaves like a polynomial within its interval of convergence. There it may be differentiated or integrated term by term; the radius stays the same, although endpoints may change. Substitution into a known series must carry the transformed convergence condition. When combining series, rewrite powers around the same center before matching coefficients. At $x=c$, all positive powers vanish, providing an immediate value check. Outside the radius, terms do not approach zero; at the boundary, use separate tests. Distinguish a formal expression from a function identity by stating where convergence and equality are proved. Algebra on series is never a license to ignore its domain.

Practice

1. State the center of $\sum n(x-2)^n$.
2. Differentiate $\sum c_nx^n$ formally.
3. What happens to radius under termwise differentiation?

Connections and next steps

• Radius of Convergence
• Interval of Convergence
• Taylor Series