Theorem: sum_range_pow
File: Mathlib/NumberTheory/Bernoulli.lean

Proof steps:
state_before: A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
⊢ ∑ k ∈ range n, ↑k ^ p = ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / (↑p + 1)
state_after: A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
hne : ∀ (m : ℕ), ↑m ! ≠ 0
⊢ ∑ k ∈ range n, ↑k ^ p = ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / (↑p + 1)
tactic: have hne : ∀ m : ℕ, (m ! : ℚ) ≠ 0 := fun m => mod_cast factorial_ne_zero m
state_before: A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
hne : ∀ (m : ℕ), ↑m ! ≠ 0
h_cauchy :
  ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => (coeff ℚ (q + 1)) (exp ℚ ^ n)) =
    PowerSeries.mk fun p => ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!
hps :
  ∑ k ∈ range n, ↑k ^ p = (∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !
⊢ ∑ k ∈ range n, ↑k ^ p = ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / (↑p + 1)
state_after: A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
hne : ∀ (m : ℕ), ↑m ! ≠ 0
h_cauchy :
  ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => (coeff ℚ (q + 1)) (exp ℚ ^ n)) =
    PowerSeries.mk fun p => ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!
hps :
  ∑ k ∈ range n, ↑k ^ p = (∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !
⊢ ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)! * ↑p ! =
    ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / (↑p + 1)
tactic: rw [hps, sum_mul]
state_before: A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
hne : ∀ (m : ℕ), ↑m ! ≠ 0
h_cauchy :
  ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => (coeff ℚ (q + 1)) (exp ℚ ^ n)) =
    PowerSeries.mk fun p => ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!
hps :
  ∑ k ∈ range n, ↑k ^ p = (∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !
⊢ ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)! * ↑p ! =
    ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / (↑p + 1)
state_after: A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
hne : ∀ (m : ℕ), ↑m ! ≠ 0
h_cauchy :
  ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => (coeff ℚ (q + 1)) (exp ℚ ^ n)) =
    PowerSeries.mk fun p => ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!
hps :
  ∑ k ∈ range n, ↑k ^ p = (∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !
x : ℕ
x✝ : x ∈ range (p + 1)
⊢ bernoulli x * ↑((p + 1).choose x) * ↑n ^ (p + 1 - x) / ↑(p + 1)! * ↑p ! =
    bernoulli x * ↑((p + 1).choose x) * ↑n ^ (p + 1 - x) / (↑p + 1)
tactic: refine sum_congr rfl fun x _ => ?_
state_before: A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
hne : ∀ (m : ℕ), ↑m ! ≠ 0
h_cauchy :
  ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => (coeff ℚ (q + 1)) (exp ℚ ^ n)) =
    PowerSeries.mk fun p => ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!
hps :
  ∑ k ∈ range n, ↑k ^ p = (∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !
x : ℕ
x✝ : x ∈ range (p + 1)
⊢ bernoulli x * ↑((p + 1).choose x) * ↑n ^ (p + 1 - x) / ↑(p + 1)! * ↑p ! =
    bernoulli x * ↑((p + 1).choose x) * ↑n ^ (p + 1 - x) / (↑p + 1)
state_after: A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
hne : ∀ (m : ℕ), ↑m ! ≠ 0
h_cauchy :
  ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => (coeff ℚ (q + 1)) (exp ℚ ^ n)) =
    PowerSeries.mk fun p => ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!
hps :
  ∑ k ∈ range n, ↑k ^ p = (∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !
x : ℕ
x✝ : x ∈ range (p + 1)
⊢ bernoulli x * ↑((p + 1).choose x) * ↑n ^ (p + 1 - x) * ↑p ! * (↑p + 1) =
    bernoulli x * ↑((p + 1).choose x) * ↑n ^ (p + 1 - x) * ((↑p + 1) * ↑p !)
tactic: field_simp [mul_right_comm _ ↑p !, ← mul_assoc _ _ ↑p !, factorial]
state_before: A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
hne : ∀ (m : ℕ), ↑m ! ≠ 0
h_cauchy :
  ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => (coeff ℚ (q + 1)) (exp ℚ ^ n)) =
    PowerSeries.mk fun p => ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!
hps :
  ∑ k ∈ range n, ↑k ^ p = (∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !
x : ℕ
x✝ : x ∈ range (p + 1)
⊢ bernoulli x * ↑((p + 1).choose x) * ↑n ^ (p + 1 - x) * ↑p ! * (↑p + 1) =
    bernoulli x * ↑((p + 1).choose x) * ↑n ^ (p + 1 - x) * ((↑p + 1) * ↑p !)
state_after: no goals
tactic: ring