Theorems · Theorem · sequences and series
summable_norm_mul_geometric_of_norm_lt_one
∀ {R : Type u_4} [inst : NormedRing R] {k : ℕ} {r : R},
‖r‖ < 1 → ∀ {u : ℕ → ℕ}, ((fun n => ↑(u n)) =O[Filter.atTop] fun n => ↑(n ^ k)) → Summable fun n => ‖↑(u n) * r ^ n‖- Defined in
- Mathlib.Analysis.SpecificLimits.Normed
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NormedRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites34
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Norm.normstatement and proof · cited by 5,413
- LE.le.transproof · cited by 3,151
- Filter.atTopstatement and proof · cited by 2,405
- LT.lt.leproof · cited by 2,189
- SummationFilter.unconditionalstatement · cited by 2,068
- Filter.univ_mem'proof · cited by 1,672
- Filter.mp_memproof · cited by 1,537
- NormedRingstatement and proof · cited by 924
- Summablestatement · cited by 778
- norm_nonnegproof · cited by 725
- Asymptotics.IsBigOstatement and proof · cited by 506
Cited by2
Results whose statement or proof uses this declaration.
- hasSum_choose_mul_geometric_of_norm_lt_one'proof · cited by 4
- summable_norm_pow_mul_geometric_of_norm_lt_oneproof · cited by 3