Alternating series test

Definition

The alternating series test determines the convergence of alternating series. A series is alternating if its terms alternate in sign.

5 Must Know Facts For Your Next Test

1. An alternating series $\sum (-1)^n a_n$ converges if $a_n$ is positive, decreasing, and approaches zero as $n$ approaches infinity.
2. The Alternating Series Test is also known as the Leibniz Criterion.
3. If an alternating series passes the test, it is only guaranteed to converge conditionally unless further tests show absolute convergence.
4. For an alternating series $\sum (-1)^n a_n$, the error estimate after truncating at the $n$-th term is less than or equal to the first omitted term $|a_{n+1}|$.
5. Alternating harmonic series like $\sum (-1)^{n+1} \frac{1}{n}$ are classic examples where the test applies.

Review Questions

• What are the three conditions that must be met for an alternating series to converge according to the Alternating Series Test?
• Explain why an alternating harmonic series converges using the Alternating Series Test.
• Describe what it means for a series to converge conditionally versus absolutely.

Related terms

Absolute Convergence:

A series $\sum a_n$ converges absolutely if $\sum |a_n|$ converges.

Conditional Convergence:

A series $\sum a_n$ converges conditionally if it converges but does not converge absolutely.

Harmonic Series:

$\sum \frac{1}{n}$, which diverges. An example of an alternating version is $\sum (-1)^{n+1} \frac{1}{n}$.