Determining Absolute or Conditional Convergence BC only

The classification is two questions in a fixed order. First, does $\sum |a_n|$ converge? If so the series is absolutely convergent and converges, and the problem is over; the order matters because that first question is about positive terms and so unlocks every test from 10.4 to 10.8, whereas the second is about mixed signs and is eligible for almost nothing but the alternating series test. If the absolute series diverges, ask whether $\sum a_n$ converges: if it does the series is conditionally convergent, and if it does not it is simply divergent.

Absolute convergence implies convergence, by comparing $0 \le a_n + |a_n| \le 2|a_n|$ and subtracting, and the converse is false, with $\sum \frac{(-1)^{n+1}}{n}$ converging to $\ln 2$ while $\sum \frac{1}{n}$ diverges. Good news transfers upward from the absolute series and never downward, and every form of the confusion is a downward transfer. The distinction is more than vocabulary: an absolutely convergent series may be reordered freely without changing its sum, while a conditionally convergent one can be rearranged to produce any value at all, because its positive and negative parts each diverge on their own. Running the flow on $\sum \frac{(-1)^{n+1}}{\sqrt{n}}$ gives divergence at step one, since $p = \frac{1}{2}$, and convergence at step two, so it is conditionally convergent, while $\sum (-1)^{n}\frac{n}{n+1}$ fails both and is divergent outright.