9.5 Alternating Series

An alternating series is a series whose terms are alternately positive and negative. It has the following forms Example: Alternating harmonic series

Alternating Series Test Example: Test the series

Remark If condition (2) of the alternating test fails, Divergence Test should work. If condition (1) of the alternating test fails, try different methods. To verify condition (1), you can show that the derivative is negative for n > k, where k is some positive constant of your choice.

Error Bounds for Alternating Series This just says that you can estimate the error on the n th partial sum by the (n+1) th term

Absolute and Conditional Convergence Example: Test the series Theorem: Definitions:

Test the series for absolute convergence: converges converges by the direct comparison test. Since converges absolutely, it converges. Example

Examples Test the series. Classify any convergent series as absolutely or conditionally convergent.