Calculus II · Unit 4B · lesson

Endpoint Testing for Power Series

Concept

Learning objectives

test each endpoint using numerical-series methods and state the complete interval of convergence.

Endpoint Testing for Power Series

Explanation

Endpoints are separate problems, not decorative brackets

After the radius is known, substituting an endpoint removes the variable and produces a numerical series. The left and right endpoints may behave differently because signs change. One may yield a harmonic series, the other an alternating harmonic series. Therefore endpoint inclusion must be recorded one endpoint at a time.

A complete interval answer shows the center, radius, and bracket choices. It also states why each endpoint is included or excluded. Merely writing an interval without tests hides the most interesting part of many power-series problems.

Decision

Do not reuse the radius calculation at an endpoint

The ratio or root limit usually equals one at the boundary and therefore says nothing. Substitute the endpoint into the original series, simplify the resulting numerical series, and choose a fresh convergence test. The two endpoints may have different outcomes.

Bridge

Each endpoint becomes its own numerical-series problem

Inside the radius, convergence is absolute; outside, divergence is guaranteed. At an endpoint, the power factor usually becomes $1$ or $(-1)^n$ , exposing a familiar numerical series. One endpoint may converge while the other diverges.

Endpoint work is therefore not a minor final checkbox. It can distinguish open, closed, and half-open intervals, and often changes absolute convergence into conditional convergence. Substitute each endpoint into the original series, simplify completely, and name the test used.

**Interior, endpoints, and exterior require different reasoning**
A number line shows an interior region decided by the radius, two endpoint test boxes, and exterior divergence.
Interior, endpoints, and exterior require different reasoning. Three-zone interval with separate endpoint result cards.

Concept

Endpoint workflow

Solve $|x-a|<R$ , then substitute $x=a-R$ and $x=a+R$ into the original series. Use ordinary convergence tests on the resulting numerical series.

Guided walkthrough

A half-open interval

For

\sum_{n=1}^{\infty}\frac{(x-1)^n}{n},

we have $|x-1|<1$ . At $x=2$ , the series is harmonic and diverges. At $x=0$ , it becomes $\sum(-1)^n/n$ , which converges. The interval is $[0,2)$ .

Worked example

One endpoint closes and the other does not

Find the interval of convergence of

\sum_{n=1}^{\infty}\frac{(x-2)^n}{n}.

The ratio test gives $|x-2|<1$ , so the candidate interval is $(1,3)$ . At $x=3$ , the series becomes $\sum1/n$ , which diverges. At $x=1$ , it becomes $\sum(-1)^n/n$ , which converges conditionally. Therefore the interval is $[1,3)$ .

Common mistake

Do not test endpoints in the simplified ratio expression

Endpoint behavior belongs to the original power series. Substituting endpoints into the ratio-test limit can only reproduce the inconclusive value one.

Interactive checku4b-endpoint_testing-01

Find the interval of convergence of $\sum_{n=1}^{\infty}(x-1)^n/n$ .

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Show hint

Radius one; test $x=0$ and $x=2$ separately.

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Exercise

Find the interval of $\sum x^n/n^2$ .

Exercise

Find the interval of $\sum (x+3)^n/[n4^n]$ .

Exercise

Give a power series with both endpoints included.

Exercise

Explain why convergence inside the radius is absolute.

After the explanation

Use the section idea

Reading lens

Find the radius from interior behavior, then test each boundary point as a separate numerical series.

Mental model

Distance from the center organizes the automatic interior and exterior behavior; endpoints remain independent decisions.

Decision

Use a ratio or root argument for the radius, convert it to an interval, and test both endpoints explicitly.

Common trap

Including or excluding both endpoints from the radius calculation alone.

Check yourself

Can you justify the radius and each endpoint with separate evidence?

Source & rights

Original instruction with traceable references.

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