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| The Ratio Test |
|---|
| To apply the ratio test to a given infinite series
we evaluate the limit
There are three possibilities:
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The proof of this test is relatively long, and as such is provided in an appendix on the Proof of the Ratio Test.
Observe Carefully: The Test Can Yield No Information
Before moving on, note that in the case that L = 1, the test yields no information. Applying the ratio test to the harmonic series
yields
Because the limit equals 1, the ratio test fails to give us any information.
But the harmonic series is not a convergent series, so in the case where L = 1, other convergence tests can be used to try to determine whether or not the series converges.
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