Theorem: NormedRing.tsum_geometric_of_norm_lt_one
File: Mathlib/Analysis/SpecificLimits/Normed.lean

tactic: rw [tsum_eq_zero_add (summable_geometric_of_norm_lt_one x h)]
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
⊢ ‖∑' (n : ℕ), x ^ n‖ ≤ ‖1‖ - 1 + (1 - ‖x‖)⁻¹
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
⊢ ‖x ^ 0 + ∑' (b : ℕ), x ^ (b + 1)‖ ≤ ‖1‖ - 1 + (1 - ‖x‖)⁻¹

tactic: simp only [_root_.pow_zero]
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
⊢ ‖x ^ 0 + ∑' (b : ℕ), x ^ (b + 1)‖ ≤ ‖1‖ - 1 + (1 - ‖x‖)⁻¹
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
⊢ ‖1 + ∑' (b : ℕ), x ^ (b + 1)‖ ≤ ‖1‖ - 1 + (1 - ‖x‖)⁻¹

tactic: refine le_trans (norm_add_le _ _) ?_
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
⊢ ‖1 + ∑' (b : ℕ), x ^ (b + 1)‖ ≤ ‖1‖ - 1 + (1 - ‖x‖)⁻¹
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
⊢ ‖1‖ + ‖∑' (b : ℕ), x ^ (b + 1)‖ ≤ ‖1‖ - 1 + (1 - ‖x‖)⁻¹

tactic: have : ‖∑' b : ℕ, (fun n ↦ x ^ (n + 1)) b‖ ≤ (1 - ‖x‖)⁻¹ - 1 := by
  refine tsum_of_norm_bounded ?_ fun b ↦ norm_pow_le' _ (Nat.succ_pos b)
  convert (hasSum_nat_add_iff' 1).mpr (hasSum_geometric_of_lt_one (norm_nonneg x) h)
  simp
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
⊢ ‖1‖ + ‖∑' (b : ℕ), x ^ (b + 1)‖ ≤ ‖1‖ - 1 + (1 - ‖x‖)⁻¹
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
this : ‖∑' (b : ℕ), (fun n => x ^ (n + 1)) b‖ ≤ (1 - ‖x‖)⁻¹ - 1
⊢ ‖1‖ + ‖∑' (b : ℕ), x ^ (b + 1)‖ ≤ ‖1‖ - 1 + (1 - ‖x‖)⁻¹

tactic: linarith
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
this : ‖∑' (b : ℕ), (fun n => x ^ (n + 1)) b‖ ≤ (1 - ‖x‖)⁻¹ - 1
⊢ ‖1‖ + ‖∑' (b : ℕ), x ^ (b + 1)‖ ≤ ‖1‖ - 1 + (1 - ‖x‖)⁻¹
state_after: no goals

tactic: refine tsum_of_norm_bounded ?_ fun b ↦ norm_pow_le' _ (Nat.succ_pos b)
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
⊢ ‖∑' (b : ℕ), (fun n => x ^ (n + 1)) b‖ ≤ (1 - ‖x‖)⁻¹ - 1
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
⊢ HasSum (fun b => ‖x‖ ^ (b + 1)) ((1 - ‖x‖)⁻¹ - 1)

tactic: convert (hasSum_nat_add_iff' 1).mpr (hasSum_geometric_of_lt_one (norm_nonneg x) h)
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
⊢ HasSum (fun b => ‖x‖ ^ (b + 1)) ((1 - ‖x‖)⁻¹ - 1)
state_after: case h.e'_6.h.e'_6
α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
⊢ 1 = ∑ i ∈ Finset.range 1, ‖x‖ ^ i

tactic: simp
state_before: case h.e'_6.h.e'_6
α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
h : ‖x‖ < 1
⊢ 1 = ∑ i ∈ Finset.range 1, ‖x‖ ^ i
state_after: no goals