Root test
For with : implies converges, implies it diverges, and is inconclusive.
Proposition 1.30 (Root Test)
If for all , then consider , and assume such that .
Then
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implies converges.
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implies diverges.
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is inconclusive.
Proof
Suppose . By the definition of limit there is an with for all , so for all , and diverges by the th term test.
Suppose . Pick with . By the definition of limit, , i.e. , for all . Since is a convergent geometric series, the comparison test gives convergence of .
Examples
converges, since . diverges, since .
Both and have : the test is inconclusive exactly when the limit equals .
Related
Stated in
- Proposition 1.30 (Root Test)§1.4 Series and Convergence Tests