Method guideCalculus IIIntermediate11 min read

Finding a power series interval of convergence

Find the radius with the Ratio or Root Test, then test both endpoints separately because the general test goes silent there.

\sum_{n=0}^{\infty}c_n(x-a)^n

Method guide

How to recognize the method, run it, and know when it is the wrong choice.

Reviewed July 11, 2026

WorkflowStart here

Solve the ratio or root inequality for |x − a| < R. Then substitute each endpoint into the original series and apply an ordinary convergence test.

Center and radius

A power series is organized around its center a. Inside a radius R it converges absolutely; outside it diverges. This all-or-nothing interior behavior is a special strength of power series.

The radius may be zero or infinite, though many course examples produce a finite positive value.

Why endpoints are separate problems

At |x − a| = R, the Ratio Test commonly returns 1 and gives no conclusion. Substitution may produce a p-series, alternating series, or another familiar form.

The two endpoints can behave differently, so test both and record brackets or parentheses accordingly.

Interval notation carries information

Parentheses mean divergence at an endpoint; brackets mean convergence. The center and radius alone do not capture endpoint behavior.

Worked exampleRadius first, endpoints second

Find the interval for Σ (x−2)ⁿ/n.

\left|\frac{a_{n+1}}{a_n}\right|\longrightarrow|x-2|<1

The Ratio Test gives radius 1.

x=3:\ \sum\frac1n\ \text{diverges}

The right endpoint is harmonic.

x=1:\ \sum\frac{(-1)^n}{n}\ \text{converges}

The left endpoint is alternating harmonic.

Result

\boxed{[1,3)}

Watch for

Common mistakes

Including both endpoints automatically after finding a radius.
Testing endpoints in the ratio inequality instead of the original series.
Forgetting that each endpoint can behave differently.

Keep

Three takeaways

Ratio or root test finds the open interior.
Endpoint tests are independent.
Report the final answer as an interval, not only a radius.