Series that Converge Absolutely but Don't Converge

Absolute Convergence in Ordered Fields

“…The constructions of this section are intended to complement those of [2]. In particular, the proof of Theorem 1 answers their Question 1 for b = 2.…”

“…[5] and [3] -it is natural to ask about convergence versus absolute convergence in an arbitrary ordered field. In a recent note, Kantrowitz and Schramm ask whether in an ordered field F , every absolutely convergent series is convergent if and only if F is sequentially complete [2,Question 3]. We will answer this question and also determine the ordered fields in which every convergent series is absolutely convergent.…”

Completeness of Ordered Fields and a Trio of Classical Series Tests

“…The "real analysis in reverse" program, as described in [1], is shedding light on the fact that so many of the theorems of real analysis are actually equivalent reformulations of the notion of completeness and hence of one another. In addition to [1], the flurry of recent articles [2][3][4][5][6][7][8][9] is dedicated to the study of abstract ordered fields, particularly the various ways in which the Archimedean and completeness properties are manifested.…”