∀ {k : ℕ},
  1 ≤ k →
    ∀ {z : ℂ},
      z ∈ UpperHalfPlane.upperHalfPlaneSet →
        iteratedDerivWithin k (fun x => ↑Real.pi * (↑Real.pi * x).cot) UpperHalfPlane.upperHalfPlaneSet z =
          (-1) ^ k * ↑k.factorial * ∑' (n : ℤ), 1 / (z + ↑n) ^ (k + 1)