If a series is absolutely convergent, it is convergent.
For example, the series
is convergent, but not
absolutely convergent, because the series
by Prop. 3.1.
Consider the series
. The corresponding
series of absolute values is
is convergent because for any
, its general term is less than or equal to
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 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.