Calculus II · Unit 4B · lesson
Using the Ratio Test on Power Series
Use the Ratio Test to determine power-series radius, simplify coefficient growth, and recognize infinite-radius examples.
Section overview
Power-series convergence and endpointsWhat this section is building
Use the Ratio Test to determine power-series radius, simplify coefficient growth, and recognize infinite-radius examples.
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
derive an inequality in from the Ratio Test and identify the radius.
Using the Ratio Test on Power Series
The Ratio Test turns coefficient growth into a distance condition
For many power series, the ratio of consecutive terms simplifies to a constant multiple of . The inequality then becomes the interior convergence condition. This is one of the most reusable calculations in Calculus II.
The ratio test is applied to the whole term, not only to coefficients. After solving the inequality, record the radius and open interval before touching endpoints. Keeping those stages separate prevents endpoint conclusions from being smuggled out of a test that is explicitly inconclusive there.
The Ratio Test turns x into a distance inequality
When a power series contains factorials or complicated coefficients, the Ratio Test usually isolates a factor of . The resulting inequality describes the open interval of absolute convergence.
The calculation finds the radius, not the complete interval. After solving the inequality, return to the original series at both endpoints. The ratio test often becomes inconclusive there precisely because its limiting ratio equals one.
Coefficient growth and distance multiply
The neighboring-term ratio often separates as
The coefficient limit sets the allowable size of . This is the series analogue of balancing coefficient growth against powers of distance.
Generic ratio pattern
For , compute
A factorial coefficient
For
the ratio is for every real . Therefore .
A factorial produces infinite radius
Find the radius of convergence of
For ,
for every real . Since the ratio limit is always below one, the series converges absolutely for all , so .
Do not replace endpoint tests with the ratio limit
If solving gives , the points where still require direct substitution into the original series.
u4b-ratio_test_for_power_series-01Find the radius of .
Your work stays on this device. No account or AI grader is used.
Show hint
The ratio includes a factor .
Attempt once to unlock the solution
Submit an answer first. The hint is available now.
Find the radius of .
Find the radius of .
Explain why the test is inconclusive at .
Derive a radius formula when .
Source & rights
Original instruction with traceable references.
BetterGrades-original; no direct adaptation declared in the verified handoff.
Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.