Ratio test counterexample

Given a sequence a₁, a₂, a₃, ... let L be the limit of the ratio of consecutive terms:

Then the series

converges if L < 1 and diverges if L > 1.

However, that's not the full story. Here is an example from Ernesto Cesaro (1859-1906) that shows the ratio test to be more subtle than it may seem at first. Let 1 < < and consider the series

The ratio a_{2n + 1} / a_2n diverges, but the sum converges.

Our statement of the ratio test above is incomplete. It should say if the limit exists and equals L, then the series converges if L < 1 and diverges if L > 1. The test is inconclusive if the limit doesn't exist, as in Cesaro's example. It's also inconclusive if the limit exists but equals 1.

Cesaro's example interweaves two convergent series, one consisting of the even terms and one consisting of the odd terms. Both converge, but the series of even terms converges faster because > .

Related post: Cesaro summation

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