The ${n}$th-term test for divergence everywhere I've seen basically states that, if the limit of the ${n}$th term as ${n}$ approaches infinity is not 0 or doesn't exist, the series diverges. However, this test does not give any precondition on the ${n}$th term (e.g. that it has to be positive), so it should be applicable to alternating series as well. So,
Shouldn't all alternating series diverge, since the limit of the nth term of the series, ${a_n =(-1)^nb_n}$ (including the ${(-1)^n}$), as n approaches infinity usually does not exist?
Won't this contradict those series that pass the Alternating Series Test (AST)? Or does the AST supersede the ${n}$th-term test in such cases (if so, why isn't this mentioned anywhere)?