Interesting question!
My professor opened up his lecture about the divergence test by introducing logic concepts. For instance, the divergence test states that
If the series a sub k converges, then the limit of a sub k (as k tends to infinity) equals zero.
If we label the premise, A, "the series a sub k converges," and the conclusion, B, "the limit of a sub k (as k tends to infinity) equals zero," then the conditional statement is in the form if A, then B.
Its human nature to make fallacious assumptions that a conclusion, B, can lead to a premise, A. that is to say, if B, then A. This assumption can only be true if the conditional statement states "A, if and only if B."
So how does logic fit into the divergence test?
Well, after lecture, I learned that the conditional statement, "If the series a sub k converges, then the limit of a sub k (as k tends to infinity) equals zero," is true, and so is the contrapositive statement (if not B, then not A). However, the inverse and converse statements of our original statement are not true.
The contrapositive statement is, "if not B, then not A":
If the limit of a sub k does NOT equal zero, then the series does NOT converge; thus we can conclude that if the limit is anything other than zero, by the divergence test, we can say that the series diverges.
Logic is a strange thing because it doesn't always agree with our intuition. Anyone that has taken a logic course (or has studied it independently) can appreciate its contents and applications.