What are the "strong" and "weak" in mathematics? If $A→B$ is $A$ strong and $B$ weak? For instance, the "strong inequality" $5<6$ implies the "weak inequality" $5≤6$. Is there anything more to it than that? I've seen the terms "stronger/weaker condition" or "stronger/weaker tests" used a few times. But I'm not sure if what I understand is entirely correct. 
 A: The 'strong' claim is stronger than the 'weak' claim in that it 'says more'. For example, if I say that 'I am going to get rich and be happy', I am making a stronger claim than when I just say 'I am going to get rich', which in turn is stronger than the claim 'I am going to get rich or be happy'
Note that $5 \le 6$ can be written as the claim $5 < 6 \lor 5 = 6$, and is therefore indeed weaker than the claim $5 < 6$.
Notice that the stronger claim is therefore always less likely to be true (and hence more likely to be false) than the weak claim. Indeed, the 'maximally' strong claim would be a contradiction: 'I am going to be rich and I am not going to be rich!'.  And a tautology like 'I am going to be rich or I am not going to be rich!' is the maximally weak claim.
In terms of information-content: the stronger claim contains more information and the weaker claim contains less information. Indeed, the maximally weak claim of 'I am going to be rich or I am not going to be rich!' ends up not saying anything at all: it has no information-content whatsoever. The maximally strong claim, the contradiction, has all information-content, which is why anything can be inferred from a contradiction.
As far as the root test vs ratio test goes: Tests are of course not the same as statements, but the same idea of 'how much something tells us' comes into play here as well: We say that the root test is stronger than the ratio test in that whenever a series 'passes' the ratio test, then it will automatically pass the root test, but the converse is not the case: something can 'pass' the root test without passing the ratio test. As a result, the root test can 'tell us more' than the ratio test: when I put a series to the ratio test, I may not gain any information as to whether that series converges or not, but if I put that same series to the root test, I may gain the information that it does converge.
We can also look at the ratio vs root test as follows:
Suppose the limit we calculate in the root test is $L_{Root}$ and the limit we calculate in the ratio test is $L_{Ratio}$. Finally, let's use $C$ for the claim 'the series converges'. Now, the two claims we have are:
$$L_{root} < 1 \rightarrow C \tag{1}$$
and
$$L_{ratio} < 1 \rightarrow C \tag{2}$$
But we know that $L_{root} \le L_{ratio}$, and thus:
$$L_{ratio} < 1 \rightarrow L_{root} < 1$$
(or again: if the series 'passes' the ratio test ($L_{ratio} < 1$) then the series 'passes' the root test ($L_{root} < 1$))
So, the antecedent of claim $(2)$ is stronger than the antecedent of claim $(1)$, but that means that the whole claim $(2)$ is weaker than claim $(1)$ (this is a valid logic principle called strengthening the antecedent). That is, if $(1)$ is true, then $(2)$ is true, making the root test the stronger test.
A: If you are a dog, then you are a mammal.  Knowing you are a dog is much more informative (stronger information) than merely knowing you are a mammal.  (Of course, you could be a horse).  So, being a dog is "stronger."  Thus in $P \Rightarrow Q$, $P$ is strong and $Q$ is weak.  
