Science:Infinite Series Module/Units/Unit 2/2.4 The Ratio Test/2.4.02 The Ratio Test

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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:

  • if L < 1, then the series converges
  • if L > 1, then the series diverges
  • if L = 1, then the test is inconclusive

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

Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} \lim_{k \rightarrow \infty} \Big| \frac{a_{k+1}}{a_k} \Big| = \lim_{k \rightarrow \infty} \Bigg| \frac{\frac{1}{k+1}}{\frac{1}{k}} \Bigg| = \lim_{k \rightarrow \infty} \frac{k}{k+1}  = 1 . \end{align} }

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.