Alternating Series

Alternating Series Test

The alternating series  converges provided  1) the terms of the series are monotonically decreasing  2) 

Remainder in Alternating Series

Let be a convergent alternating series with terms that are non increasing magnitude. Let be the remainder in approximating the value of the series by the sum of its first n terms. Then . In other words, the remainder is less than or ewaul to the magnitude of the first neglected term.

Absolute and Coditional Convergence

If converges, then converges absolutely. If diverges and converges, then converges conditionally.