Theorems · Theorem · sequences and series

Summable.tsum_eq_add_tsum_ite

∀ {α : Type u_1} {β : Type u_2} [inst : AddCommGroup α] [inst_1 : TopologicalSpace α] [IsTopologicalAddGroup α]
  {f : β → α} [T2Space α] [inst_4 : DecidableEq β],
  Summable f → ∀ (b : β), ∑' (n : β), f n = f b + ∑' (n : β), if n = b then 0 else f n

Let f : β → α be a summable function and let b ∈ β be an index. Lemma tsum_eq_add_tsum_ite writes Σ' n, f n as f b plus the sum of the remaining terms.

Defined in

Mathlib.Topology.Algebra.InfiniteSum.Group

Cited by

1 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupTopologicalSpaceIsTopologicalAddGroupT2SpaceDecidableEq