Theorems · Theorem · sequences and series
ENNReal.tsum_geometric
∀ (r : ENNReal), ∑' (n : ℕ), r ^ n = (1 - r)⁻¹
The series pow r converges to (1-r)⁻¹. For r < 1 the RHS is a finite number,
and for 1 ≤ r the RHS equals ∞.
- Defined in
- Mathlib.Analysis.SpecificLimits.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 137 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites30
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- ENNRealstatement and proof · cited by 9,879
- Top.topproof · cited by 9,680
- NNRealproof · cited by 4,310
- Nat.cast_oneproof · cited by 2,501
- Finset.cardproof · cited by 2,327
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- Finset.rangeproof · cited by 1,341
- ENNReal.ofNNRealproof · cited by 1,279
- tsumstatement and proof · cited by 1,148
- ne_of_gtproof · cited by 637
- lt_of_lt_of_leproof · cited by 438
- Finset.sum_constproof · cited by 254
Cited by8
Results whose statement or proof uses this declaration.
- MeasureTheory.TendstoInMeasure.exists_seq_tendsto_aeproof · cited by 5
- edist_le_of_edist_le_geometric_of_tendstoproof · cited by 3
- cauchySeq_of_edist_le_geometricproof · cited by 2
- MeasureTheory.Measure.exists_null_set_measure_lt_of_disjointproof · cited by 1
- ENNReal.tsum_geometric_add_oneproof · cited by 1
- ENNReal.tsum_geometric_twoproof · cited by 1
- tsum_geometric_lt_topproof · cited by 1
- tsum_geometric_encode_lt_topproof · cited by 0