Calculus II · Unit 4B · lesson
Endpoint Testing for Power Series
Test power-series endpoints independently with ordinary series methods and write the final interval with correct inclusion.
Section overview
Power-series convergence and endpointsWhat this section is building
Test power-series endpoints independently with ordinary series methods and write the final interval with correct inclusion.
Distance from the center organizes the automatic interior and exterior behavior; endpoints remain independent decisions.
Use a ratio or root argument for the radius, convert it to an interval, and test both endpoints explicitly.
Including or excluding both endpoints from the radius calculation alone.
Learning objectives
test each endpoint using numerical-series methods and state the complete interval of convergence.
Endpoint Testing for Power Series
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.
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.
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 or , 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.
Read this graph as text
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. Three-zone interval with separate endpoint result cards. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in interior, endpoints, and exterior require different reasoning; color is never the only cue.
Why it matters: Three-zone interval with separate endpoint result cards.
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.
Endpoint workflow
Solve , then substitute and into the original series. Use ordinary convergence tests on the resulting numerical series.
A half-open interval
For
we have . At , the series is harmonic and diverges. At , it becomes , which converges. The interval is .
One endpoint closes and the other does not
Find the interval of convergence of
The ratio test gives , so the candidate interval is . At , the series becomes , which diverges. At , it becomes , which converges conditionally. Therefore the interval is .
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.
u4b-endpoint_testing-01Find the interval of convergence of .
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Radius one; test and separately.
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Find the interval of .
Find the interval of .
Give a power series with both endpoints included.
Explain why convergence inside the radius is absolute.
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