Calculus II · Unit 4A · lesson
The Root Test
Extract the exponential base hidden in nth powers, apply the Root Test, and interpret limits below, above, or equal to one.
Section overview
Alternating, absolute, ratio, and root testsWhat this section is building
Extract the exponential base hidden in nth powers, apply the Root Test, and interpret limits below, above, or equal to one.
Absolute convergence controls magnitude strongly enough to imply convergence; conditional convergence relies on cancellation.
Test absolute values first when practical, use the alternating-series hypotheses explicitly, and reserve ratio or root tests for matching algebraic structure.
Calling any alternating-looking series convergent or treating a ratio/root limit of one as a verdict.
Learning objectives
use nth roots to detect terms raised to the nth power and compare the Root Test with the Ratio Test.
The Root Test
nth powers reveal themselves under nth roots
When an entire expression is raised to the th power, taking an th root removes that outer exponent and exposes the effective geometric ratio. The Root Test is therefore natural for terms such as and less natural for plain rational functions.
The conclusion mirrors the Ratio Test. A limit below one gives absolute convergence; above one gives divergence; equal to one is inconclusive. In more advanced analysis, the test is often stated with a limit superior because the ordinary limit need not exist. Calculus examples are usually designed so the simpler limit exists.
Extract the effective exponential base
The Root Test asks for the long-run base hidden inside by examining . It is especially effective when the entire term is raised to the th power or contains several exponential factors.
Its conclusions parallel the Ratio Test: a limit below one gives absolute convergence, a limit above one gives divergence, and a limit equal to one is inconclusive. Simplify the th root before taking the limit; that is where the method earns its keep.
The extracted base controls the whole term
If the root limit is below one, choose above the limit. Eventually , hence , and geometric comparison proves absolute convergence.
Root Test
Let
If , the series converges absolutely. If , it diverges. If , the test is inconclusive.
An nth-power series
For
we have
Therefore the series converges.
An nth power reveals its own base
Test
Then
Since , the series converges absolutely by the Root Test.
Take the root of the whole magnitude
The test uses . Applying the root to only one factor can destroy the expression’s effective base.
u4a-root_test-01Find the Root-Test limit for .
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Show hint
The nth root cancels the outer power.
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Apply the Root Test to .
Explain why it is inconclusive for .
Compare the algebra required by root and ratio tests for .
State the limit-superior version informally.
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