Interval of Convergence

The central idea

A power series $\sum c_n(x-a)^n$ converges for $|x-a|<R$ and diverges for $|x-a|>R$, for some radius $R\in[0,\infty]$. The interval of convergence adds the real endpoints $a-R$ and $a+R$ only when separate substitution tests show convergence there.

Definitions, hypotheses, and notation

At every interior point a power series converges absolutely; conditional convergence can occur only at boundary points. The interval may be open, closed, or half-open, but it always has the radius-determined center when $0<R<\infty$. $R=0$ leaves only the center, while $R=\infty$ gives all real numbers and no finite endpoints to test.

Differentiating or integrating a power series preserves its radius but may change endpoint behavior because coefficients acquire factors or divisors involving $n$. Endpoint tests must therefore be redone for the transformed series.

Conceptual meaning

Inside the radius, exponential decay in $(x-a)^n$ dominates coefficient growth; outside, terms fail to be controlled. At the boundary that decisive ratio becomes one, so harmonic, alternating, or other endpoint-specific behavior takes over.

A dependable method and decision rule

1. Use the ratio or root test to solve the strict inequality for $x$.
2. Identify the center and provisional open interval.
3. Substitute the left endpoint into the original series.
4. Substitute the right endpoint independently.
5. Combine the results using correct brackets and parentheses, with endpoint classifications if useful.

Fully worked example

Common mistakes and why they fail