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

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.

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

- 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.