Consider a series  ∑ a_k  with positive terms. We will consider the following more general versions of the Root test and the Ratio test.

"Limsup" version:
Define

(RoLC): If ϱ < 1 then the series converges.
(RoLD): If ϱ > 1 then the series diverges.
(RaLC): If λ < 1 then the series converges.
(RaLD): If λ* > 1 then the series diverges.

Note on the shortcuts: "Ro" for Root, "Ra" for Ratio, "L" for limit test, "C" for convergent part, "D" for divergent part.

Before we pass to the "inequality" versions, we address one interesting anomaly. Note that in the case of the Root test, we only have problems with limsup being equal to 1. On the other hand, in the Ratio test we have no information about series for which limsup is at least 1 and liminf at most 1, which is a huge family, since for a typical sequence its limsup is larger than liminf. Thus the "uncertain region" of the Ratio test is much larger than for the Root test. Could we replace the "liminf" in the last statement by "limsup"? Definitely not. It is easy to cook up a convergent series for which limsup would be greater than 1, for instance this problem in Solved Problems - Testing convergence has even λ = ∞ and yet that series converges. So the statements are not as nice as we would like, but it is the best we can do with limit versions.

Now we recall the general inequality versions.

"Inequality" version:
(RoIC): If there is N and ϱ < 1 such that  (a_k)^1/k ≤ ϱ  for all k > N, then the series converges.
(RoID): If  (a_k)^1/k ≥ 1  for infinitely many k, then the series diverges.
(RaIC): If there is N and λ < 1 such that  a_k+1/a_k ≤ λ  for all k > N, then the series converges.
(RaID): If there is N so that  a_k+1/a_k ≥ 1  for all k > N, then the series diverges.

Again, there is a difference in "divergent" statements between the Root test and the Ratio test. Above we quoted a solved problem, the same series shows that a convergent series may have infinitely many ratios a_k+1/a_k that are greater than 1, so again this is the best we can do, we do have to require "all" of them to be large in (RaID).

Now we will pass to the interesting part, we will compare the "L" versions with the "I" versions to see which one is more general. To this purpose we will compare assumptions of the statements above.

1. The assumptions of (RoLC) and (RaLC) are exactly equivalent to the assumptions of (RoIC) and (RaIC), respectively. In other words, if some series satisfies assumptions of (RoLC), then it also satisfies assumptions of (RoIC), and vice versa, so these two tests give convergence for the same series; the same is true for convergence parts of (ReLC) and (ReIC). Thus for convergent cases the limit versions and the inequality versions are of the same "strength", none is better than the other, they are just two different ways of stating the same thing.

2. The assumptions of (RoLD) and (RaLD) are stronger than the assumptions of (RoIC) and (RaIC), respectively. This means that if some series satisfies, say, the assumption of (RoLD), then it also automatically satisfies the assumption of (RoID), therefore we can use the latter test instead of the former test. However, not every series that satisfies the assumption of (RoID) also satisfies the assumption of (RoLD). Precisely, knowing that  (a_k)^1/k ≥ 1  for infinitely many k means that the limit superior can be greater than 1 but also equal to 1. For such series we are in situation that the "inequality" version of the Root test can be used, but the "limsup" version may come up indecisive.

Similarly, if a series satisfies the assumption of (RaLD), then it also must satisfy the assumption of (RaID), but it does not work the other way around, since the case of liminf equal to 1 may appear. The conclusion therefore is that when it comes to the "divergent" statements, the "inequality" versions of the two tests are stronger, more universal. In other words, every time the limit versions lead to some conclusion we could also use the inequality versions, but there are situations when only the latter can help and the limit ones fail.