Definition
Alternating Series
An infinite series
is called alternating if its terms alternate in sign. This is typically written in the form: where
for all .
Convergence
An alternating series does not necessarily converge. The most common tool for testing the convergence of such series is the Leibniz Criterion.
Leibniz's Theorem
The alternating series
converges if:
- The sequence
is monotonically decreasing ( ). - The sequence
is a zero sequence ( ).
Examples
- Alternating Harmonic Series:
. This series is conditionally convergent. - Geometric Alternating Series:
. This series is absolutely convergent.