∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [inst_2 : IsStrictOrderedRing α]
  [inst_3 : Ring β] {abv : β → α} [IsAbsoluteValue abv] [Archimedean α] [Nontrivial β] (x : β),
  abv x < 1 → IsCauSeq abv fun n => ∑ m ∈ Finset.range n, x ^ m