Topic 10.7 Alternating Series Test for Convergence: use the alternating series test (terms decreasing in magnitude to zero) to conclude convergence (BC).

What this topic is asking

The College Board (Topic 10.7, BC only) gives the alternating series test, the one convergence test built for series whose terms switch sign. Alternating series can converge even when the corresponding all-positive series diverges, because the alternating signs let positive and negative contributions cancel.

The test and its two conditions

Why alternating signs rescue convergence

The mechanism is that the partial sums oscillate and close in. Because the terms alternate in sign and shrink in magnitude, each partial sum overshoots the limit, then the next undershoots by less, then overshoots by less still, trapping the limit between consecutive partial sums in ever-tighter brackets. This "shrinking zigzag" forces the partial sums to converge. The all-positive version has no cancellation, so its partial sums can march off to infinity (as the harmonic series does). This is exactly why $\sum\frac{(-1)^{n+1}}{n}$ converges to $\ln 2$ while $\sum\frac{1}{n}$ diverges: same magnitudes, but the alternating signs allow the cancellation that the positive series lacks. The same oscillation gives the error bound of Topic 10.8.

A worked convergence

Checking "decreasing" carefully

The decreasing condition is where alternating-series questions are won or lost on the free-response section. You must show $b_{n+1} \le b_n$, and the cleanest justification is usually to view $b_n = f(n)$ and show $f'(x) < 0$ for large $x$, as in the worked example. Sometimes a direct algebraic comparison of $b_{n+1}$ and $b_n$ is simpler. The condition only needs to hold eventually (for $n$ beyond some point), since finitely many terms cannot affect convergence. A complete answer states both conditions explicitly and justifies each; merely asserting "the terms decrease to zero" without support does not earn full credit.

What the test does not tell you

The alternating series test proves convergence of the alternating series, but it does not tell you whether the convergence is absolute or conditional, and it gives no value for the sum. Whether $\sum a_n$ converges absolutely (meaning $\sum |a_n|$ also converges) is a separate question handled in Topic 10.9; the alternating harmonic series converges by this test but only conditionally, since $\sum\frac{1}{n}$ diverges. Also, if either condition fails, the test is inconclusive about convergence by this route, though a failure of condition 2 ($b_n\not\to 0$) actually means the original series diverges by the $n$-th term test. Keeping straight what the test does and does not establish prevents overclaiming on the exam.