Absolute convergence
converges absolutely when converges. Absolute convergence implies convergence, but not conversely.
Definition 1.41 (Absolute Convergence)
Absolute convergence implies convergence
Lemma 1.42
If converges absolutely, then it converges.
Proof
Let and . Since converges, it is Cauchy: for ,
Thus is Cauchy, and by completeness of and it converges.
Conditional convergence
The converse fails: converges by the alternating series test while diverges, so it converges without converging absolutely. Series that converge but not absolutely are called conditionally convergent.
Conditional convergence is a strictly weaker notion: such series can behave badly under rearrangements, whereas absolutely convergent series are insensitive to the order of summation.
Related
Stated in
- Definition 1.41 (Absolute Convergence)§1.4 Series and Convergence Tests
- Lemma 1.42§1.4 Series and Convergence Tests