Theorems · Theorem · sequences and series

Antitone.alternating_series_le_tendsto

∀ {E : Type u_4} [inst : Ring E] [inst_1 : PartialOrder E] [IsOrderedRing E] [inst_3 : TopologicalSpace E]
  [OrderClosedTopology E] {l : E} {f : ℕ → E},
  Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) Filter.atTop (nhds l) →
    Antitone f → ∀ (k : ℕ), ∑ i ∈ Finset.range (2 * k), (-1) ^ i * f i ≤ l

Partial sums of an alternating antitone series with an even number of terms provide lower bounds on the limit.

Defined in

Mathlib.Analysis.SpecificLimits.Normed

Cited by

1 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingPartialOrderIsOrderedRingTopologicalSpaceOrderClosedTopology