Given a sequence *a*_{1}, *a*_{2}, *a*_{3}, … 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 Cesàro (1859–1906) that shows the ratio test to be more subtle than it may seem at first. Let 1 < α < β 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 Cesàro’s example. It’s also inconclusive if the limit exists but equals 1.

Cesàro’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 β > α.

**Related post**: Cesàro summation

This doesn’t look especially subtle to me. The ratio test asks for the limit of a_{n+1} / a_n . Then the “counterexample” wants to talk about the limit of a completely different ratio. And the limit of that ratio turns out to be irrelevant to the ratio test. And…?

The original statement (“let L = lim_{n \to \infty} | a_{n+1} / a_n |; the series sum(a_n) converges if L < 1 and diverges if L > 1”) isn’t incomplete at all; when L does not exist, we cannot claim that L < 1 or that L > 1.

Typo: @1st sentence: missing ellipsis after “a1, a2, a3,”

I agree with Michael Watts, the existence of the limit is implicit since one set $L = \lim_{n\to\infty} $…

The ratio test has a different form in German math tradition [w/o using limits and slightly stronger]

https://de.wikipedia.org/wiki/Quotientenkriterium

The Cesáro example disproving the converse of the test’s implication still applies.

Hint for WW and JJAG: The shorthand “L = lim …” usually does not imply the existence of the limit. See e.g.

https://en.wikipedia.org/wiki/Ratio_test

The “real” (as opposed to Calculus II) Ratio Test uses limsup and liminf, not ordinary limits. My guess is it would stop this from being a useful example.