AP Calculus BC

Alternating Series Test vs Absolute Convergence

The alternating series test proves that the alternating series itself converges, and says nothing about the series of absolute values. Absolute convergence is the stronger conclusion, and you reach it by running a separate test on the absolute values, usually a comparison, a p-series, or the ratio test.

Alternating series test

Use when: The signs alternate, the term sizes are eventually decreasing, and the terms go to zero, and you only need to know whether the series converges.

Absolute convergence

Use when: You need the stronger conclusion, or you want to classify the convergence as absolute or conditional, so you strip the signs and test the sizes on their own.

Side by side

	Alternating series test	Absolute convergence
What it settles	Whether $\sum (-1)^n b_n$ converges	Whether $\sum \lvert a_n \rvert$ converges
What you check	$b_n$ eventually decreasing and $\lim_{n \to \infty} b_n = 0$	Any test that works on positive terms
Strength of the conclusion	Weaker; the convergence may be conditional	Stronger; it forces the original series to converge too
Example it handles	$\sum \frac{(-1)^n}{n}$ , which converges conditionally	$\sum \frac{(-1)^n}{n^2}$ , which converges absolutely
Common trap	Checking only that the terms go to zero	Assuming the alternating series test already proved it

Classifying a series with mixed signs takes two passes, not one. First the alternating series test on the series as written, then a fresh test on $\sum \lvert a_n \rvert$ . Converges both times means absolute convergence; converges then diverges means conditional convergence. The alternating harmonic series is the standard case: it passes the alternating series test, but its absolute values are the harmonic series, which diverges.

The ratio and root tests are stated with absolute values built in, so whenever either concludes convergence it has already delivered absolute convergence and no second pass is needed. That is why they are the default openers on a power series.

The error bound rides on the same three conditions

When the alternating series test conditions hold, the error after $n$ terms is no larger than the size of the first omitted term. Absolute convergence gives you no such bound on its own, so error questions send you back to the alternating series test.

Frequently asked questions

If the alternating series test passes, is the series absolutely convergent?

Not necessarily. The test only certifies the series as written. Test $\sum \lvert a_n \rvert$ separately: if that diverges the convergence is conditional, as it is for $\sum \frac{(-1)^n}{n}$ .

Can the alternating series test prove divergence?

No. If the terms fail to approach zero, it is the nth term test that gives divergence. If the terms shrink to zero but are not eventually decreasing, the alternating series test is simply inconclusive and you need another tool.

Which order should I test in?

Test the absolute values first when they look like a p-series or a geometric series, since absolute convergence settles everything at once. Fall back on the alternating series test when the absolute values diverge.

In the CED: Unit 10: Infinite Sequences and Series (BC)