Ratio test
For with : implies converges, implies it diverges, and is inconclusive.
Proposition 1.32 (Ratio 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.
Examples
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Both (divergent) and (convergent) have ratio limit , so the test is inconclusive for both.
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converges, since
The same conclusion follows from the root test, because , where by L’Hospital’s rule.
If the ratio test is inconclusive then so is the root test, but not conversely: for
the ratio limit does not exist while the root limit equals .
Related
Stated in
- Proposition 1.32 (Ratio Test)§1.4 Series and Convergence Tests