Plain-language statement
Summability of (m+1)^k * exp(-2πm) via comparison with shifted sum.
Exact Lean statement
lemma summable_pow_shift (k : ℕ) :
Summable fun m : ℕ => (m + 1 : ℝ) ^ k * rexp (-2 * π * m)Formal artifact
Lean source
lemma summable_pow_shift (k : ℕ) : Summable fun m : ℕ => (m + 1 : ℝ) ^ k * rexp (-2 * π * m) := by have h := Real.summable_pow_mul_exp_neg_nat_mul k (by positivity : 0 < 2 * π) have h_eq : ∀ m : ℕ, (m + 1 : ℝ) ^ k * rexp (-2 * π * m) = rexp (2 * π) * ((m + 1) ^ k * rexp (-2 * π * (m + 1))) := fun m => by have : rexp (-2 * π * m) = rexp (2 * π) * rexp (-2 * π * (m + 1)) := by rw [← Real.exp_add] ring_nf rw [this] ring simp_rw [h_eq] apply Summable.mul_left refine (h.comp_injective Nat.succ_injective).congr ?_ intro i simp [Function.comp_apply, Nat.succ_eq_add_one]- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/EisensteinAsymptotics.lean:229-243
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Plain-language statement
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Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
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antiSerreDerPos
Plain-language statement
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Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
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closedBall_center_subset_upperHalfPlane
Plain-language statement
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Source project: Sphere Packing in Dimension 8
Person-level attribution pending.