# Absolute convergence and conditional convergence.

Definition 9.5.1   A series is absolutely convergent if the series is convergent.

Proposition 9.5.2   If a series is absolutely convergent, it is convergent.

Example 9.5.3   Consider the series . The corresponding series of absolute values is is convergent because for any , its general term is less than or equal to (v.s. 3.3).

Thus, the given series is absolutely convergent, and is convergent.

For example, the series is convergent, but not absolutely convergent, because the series is divergent, by Prop. 3.1.

A convergent series which is convergent, but not absolutely convergent is conditionally convergent.

For example, the alternating harmonic series is convergent (v.s. 4.3), but not absolutely convergent (v.s. 1.9), thus it is conditionally convergent.

Theorem 9.5.4   Let and be two series, where the sequence is a rearrangement of the sequence .

If is absolutely convergent, then is absolutely convergent, and the two series have the same sum.

Remark 9.5.5   If the series is conditionally convergent, we can reaarange its terms so that it will show any predecided behaviour: divergence or convergence (with any limit we want!). For example, consider the alternating harmonic sequence.
• With the following rearrangement of terms

the series is divergent.
• Start with 1, substract , then add , then add consecutive negative terms so that the sum will be less than 1, then add consecutive positive terms until the sum is 1 or more, and iterate this process. You get a convergent series whose sum is 1.

Noah Dana-Picard 2007-12-28