| import Mathlib.Analysis.SpecificLimits.Basic | |
| namespace OAI | |
| namespace Yau.Analysis | |
| open Filter | |
| open scoped Topology | |
| noncomputable section | |
| theorem hasSum_scaled_geometric_shift (r : ℝ) (m : ℕ) : | |
| HasSum (fun n : ℕ ↦ r*(1/2:ℝ)^(n+m)) (r*(1/2:ℝ)^m*2) := by | |
| simpa only [pow_add,mul_assoc,mul_left_comm,mul_comm] using | |
| hasSum_geometric_two.mul_left (r*(1/2:ℝ)^m) | |
| theorem tsum_scaled_geometric_two (r : ℝ) : | |
| (∑' n : ℕ, r*(1/2:ℝ)^(n+2)) = r/2 := by | |
| convert (hasSum_scaled_geometric_shift r 2).tsum_eq using 1 | |
| ring | |
| theorem tsum_scaled_geometric_three (r : ℝ) : | |
| (∑' n : ℕ, r*(1/2:ℝ)^(n+3)) = r/4 := by | |
| convert (hasSum_scaled_geometric_shift r 3).tsum_eq using 1 | |
| ring | |
| end | |
| end Yau.Analysis | |
| end OAI | |