Two conditions, and a series that converges only because it alternates BC only

For a series $\sum (-1)^{n+1} b_n$ with every $b_n > 0$, the series converges if

1. $b_{n+1} \le b_n$ for all large $n$, that is the absolute values are eventually decreasing, and
2. $\lim_{n \to \infty} b_n = 0$.

Both are required, and they are not the same statement. A sequence can tend to zero without decreasing: $b_n$ given by $\frac{1}{2}, 1, \frac{1}{4}, \frac{1}{3}, \frac{1}{6}, \frac{1}{5}, \dots$ tends to zero while rising at every other step, and an alternating series built from it need not converge.

Why the two together suffice is visible rather than technical. Each partial sum steps forward by $b_n$ and then back by $b_{n+1} \le b_n$, so the odd partial sums decrease, the even ones increase, and every odd one stays above every even one. The two interleaved sequences squeeze towards each other, and since the gap between consecutive partial sums is exactly $b_n \to 0$, they meet. The decrease condition is what makes the squeeze, and the limit condition is what closes it.

Take $b_n = \frac{1}{n}$. It decreases and tends to zero, so

$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$$

converges, and its sum is $\ln 2 \approx 0.6931$. Meanwhile $\sum \frac{1}{n}$ diverges, as 10.5 proved. The same terms, differing only in sign, give a finite sum in one arrangement and no sum at all in the other.

That is worth pausing on, because it means the alternation is doing real work rather than being a decoration. The partial sums of the harmonic series climb forever; the partial sums here climb, fall back a little less far, climb a little less far, and converge. The figure shows the oscillation narrowing, and the narrowing is the convergence.

A series that converges while its absolute values do not is called conditionally convergent. One whose absolute values also converge is absolutely convergent. The alternating harmonic series is the standard example of the first, and $\sum \frac{(-1)^{n+1}}{n^{2}}$ of the second, since $\sum \frac{1}{n^{2}}$ converges too. Topic 10.9 is about telling them apart and about why the distinction matters.

Three limits on the conclusion, each of which the code records as an over-reading:

1. Failing the test is not divergence. If $b_n$ does not decrease, the test does not apply and the series may still converge by some other route. Only the $b_n \not\to 0$ case gives divergence, and that is really the nth term test from 10.3 doing the work.
2. The test says nothing about $\sum |a_n|$. It certifies the alternating series alone. Concluding that the absolute series converges too is exactly the CA10 confusion.
3. It gives no value. Like every test in this unit apart from the geometric formula, it decides existence. That $\ln 2$ is the sum here comes from elsewhere.

It does, however, come with something no other test provides: an error bound. Because the partial sums bracket the limit, stopping after $n$ terms leaves an error smaller than the first term you left out. That is 10.10's subject, and it is available only because of the squeeze described above.

The limit condition is usually obvious. The decrease condition takes a moment, and there are three ways to establish it:

1. By inspection, when $b_n$ is visibly a decreasing expression such as $\frac{1}{n^{2}+1}$.
2. By ratio, showing $\frac{b_{n+1}}{b_n} \le 1$.
3. By derivative, showing the corresponding $f(x)$ has $f'(x) \le 0$ eventually. This is the reliable move when $b_n$ is a quotient with competing parts, such as $\frac{n}{n^{2}+1}$ or $\frac{\ln n}{n}$.

Note the word eventually again. $\frac{\ln n}{n}$ increases up to $n = 2$ and decreases after, and that is fine: the test needs the condition only from some index onward, for the same reason 10.4's hypotheses did.

One more caution. The signs have to genuinely alternate, term by term. $\sum \frac{\cos n}{n}$ changes sign irregularly rather than alternately, so this test does not apply to it at all, and neither does any comparison test, since the terms are not positive.