Alternating Series Test for Convergence BC only

For $\sum(-1)^{n+1}b_n$ with $b_n > 0$, convergence follows from two conditions: the $b_n$ are eventually decreasing and they tend to zero. Both are needed and they differ, since a sequence can tend to zero while rising at every other step. The mechanism is a squeeze: each partial sum steps forward by $b_n$ and back by $b_{n+1} \le b_n$, so the odd partial sums fall, the even ones rise, every odd one stays above every even one, and the gap between consecutive partial sums is exactly $b_n \to 0$. The decrease makes the squeeze and the limit closes it.

With $b_n = \frac{1}{n}$ this gives the alternating harmonic series converging to $\ln 2 \approx 0.6931$, while $\sum \frac{1}{n}$ diverges: the same terms, differing only in sign, with a finite sum in one arrangement and none in the other. A series that converges while its absolute values do not is conditionally convergent, and one whose absolute values converge too is absolutely convergent, the distinction 10.9 develops. The test certifies only the alternating series, says nothing about $\sum|a_n|$, gives no value, and does not prove divergence when it fails to apply, though it does supply the error bound of 10.10. The decrease condition is checked by inspection, by ratio, or by derivative, and only needs to hold eventually, while irregular sign changes such as $\frac{\cos n}{n}$ are outside the test entirely.