Alternating series

Definition

An alternating series is a series whose terms alternate in sign. It can be expressed as $\sum (-1)^n a_n$ or $\sum (-1)^{n+1} a_n$, where $a_n$ is a sequence of positive terms.

5 Must Know Facts For Your Next Test

1. An alternating series converges if the absolute value of its terms decreases monotonically and approaches zero.
2. The Alternating Series Test, also known as the Leibniz Test, is used to determine the convergence of an alternating series.
3. If an alternating series satisfies the conditions of the Alternating Series Test, its remainder after $n$ terms is less than or equal to the first omitted term in absolute value.
4. The convergence of an alternating series does not imply absolute convergence; some alternating series converge conditionally.
5. Examples of well-known alternating series include the alternating harmonic series and the Taylor series for functions like $\sin(x)$.

Review Questions

• What are the conditions for an alternating series to converge?
• Describe how you would estimate the error when truncating an alternating series after $n$ terms.
• Provide an example of a conditionally convergent alternating series.

Related terms

Absolute Convergence:

A series $\sum a_n$ is said to be absolutely convergent if $\sum |a_n|$ converges.

Conditional Convergence:

A series that converges but does not converge absolutely.

Leibniz Test:

Another name for the Alternating Series Test; it determines if an alternating series converges based on specific criteria.