Theorems · Theorem · sequences and series
HasSum.tendsto_sum_nat
∀ {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : TopologicalSpace M] {m : M} {f : ℕ → M},
HasSum f m → Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop (nhds m)If f : ℕ → M has sum m, then the partial sums ∑ i ∈ range n, f i converge
to m.
- Cited by
- 23 results in Mathlib
- Foundations
- Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement and proof · cited by 24,529
- AddCommMonoidstatement and proof · cited by 12,281
- nhdsstatement · cited by 5,554
- Finset.sumstatement · cited by 5,195
- Filter.Tendstostatement · cited by 3,814
- Filter.atTopstatement · cited by 2,405
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- Finset.rangestatement · cited by 1,341
- Filter.Tendsto.compproof · cited by 560
- HasSumstatement and proof · cited by 518
- Filter.tendsto_finset_rangeproof · cited by 15
Cited by23
Results whose statement or proof uses this declaration.
- Complex.exp_eq_exp_ℂproof · cited by 8
- hasSum_iff_tendsto_nat_of_summable_normproof · cited by 4
- MeasureTheory.VectorMeasure.tendsto_vectorMeasure_iUnion_atTop_natproof · cited by 3
- Summable.tsum_le_of_sum_range_leproof · cited by 3
- Summable.tsum_pow_mul_one_subproof · cited by 3
- Summable.one_sub_mul_tsum_powproof · cited by 3
- Summable.hasSum_iff_tendsto_natproof · cited by 3
- Summable.tendsto_sum_tsum_natproof · cited by 3
- cauchySeq_of_edist_le_of_summableproof · cited by 2
- ContinuousLinearMap.exists_preimage_norm_leproof · cited by 2
- ZetaAsymptotics.termTSum_of_ltproof · cited by 2
- tendsto_sum_nat_addproof · cited by 2