Section Summary: 12.6

Definitions

Absolutely convergent: A series is called absolutely convergent if the series of absolute values converges.

Conditionally convergent: A series is called conditionally convergent if the series of is convergent but not absolutely convergent.

Theorems

If a series is absolutely convergent, then it is convergent.

The ratio test:

1. If

   

   then the series is absolutely convergent.

2. If

   

   (or ) then the series is divergent.

The root test:

1. If

   

   then the series is absolutely convergent.

2. If

   

   (or ) then the series is divergent.

Properties, Hints, etc.

One of the really freaky things about conditionally convergent series is that re-arrangement of terms results in any sum one desires. (This is not the case with absolutely convergent series).

Summary

The difference in the ratio test in this section is that it is ``self-referrential'': we compare terms of with other terms (rather than with some other series).