I have no idea at all what you mean by "test of reason"!
Is this a language problem? Do you mean "ratio test"?
(Hey, makes sense- ratio- rational- reason!)
The ratio test for convergence of a series of positive numbers,
, is to look at the limit of the ration
. If that limit is less than 1, the series converges. If it is greater than 1, the series converges. If it equal to one, it does not tell us if the series converges or not.
So one situation in which the ratio test does not work (I am not sure I would say "does not apply") is when that limit of the ratio is 1.
On the face of it, it looks like the ratio test can only apply to series of
positive numbers, but if you have a series of all negative numbers, you could factor out -1 and apply it to the new series with positive numbers which converges if and only if the original series does.
If you have a mixture of positive and negative numbers, you cannot apply the ratio test- although you could take the absolute value and determine
absolute convergence. If a series converges absolutely, then it converges but there exist many series which converge, but not absolutely.