The ratio test is a very useful test for convergence, whether of sequences or of series. As usual, it comes in many forms: here is one.
Note here that we are using the standard notions of convergence and divergence for sequences and series of real (or complex) numbers here.
The ratio test for sequences and series
Let be a sequence of non-zero real (or complex) numbers, and suppose that as . (Here you may allow if you wish.)
- If , then the series is absolutely convergent, and (hence) as .
- If , then the series is divergent. Moreover, as , and (hence) the sequence is divergent.
- If then the ratio test is inconclusive, and you need to try another test.
In my FAQ document for Mathematical Analysis, I recommend using the ratio test if your terms involve things like powers of constants (e.g. ) or factorials. For example, the ratio test quickly shows you the divergence of the sequence when the terms are or , and the convergence to zero of the sequence when .
If, instead, the terms are simply rational functions of , then the ratio test is always inconclusive! For example, the ratio test does not help you when is , or , or . For all three of these examples, the limiting ratio exists, but is equal to
Of course, the limiting ratio need not exist at all. (You can still make further progress using and instead.) So that is another obvious way for the ratio test, as stated above, to be inconclusive.
But one rather unusual way for this version of the ratio test to be inconclusive is, I would argue, the following case.
Suppose that . Common sense tells us that this sequence converges to zero very fast. This is, indeed, true! But what happens if we try to use the ratio test in the form we have stated it above to prove that the sequence converges to ?
Well, we have,
Is this a problem? Well, if you are trying to investigate the limiting ratio, you now find yourself back where you started! That is, you have a sequence which you think converges to zero, and you think you might prove it using the ratio test …
Of course, if you use a suitable alternative version of the ratio test, the fact that the ratios are always less than is immediately conclusive.
Note that the same ratio test issue arises if you take .