bookshelf/Bookshelf/Real/Sequence/Geometric.lean

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import Mathlib.Data.Real.Basic
2023-05-04 22:37:54 +00:00
/-! # Bookshelf.Real.Sequence.Geometric
A characterization of a geometric sequence, i.e. a sequence with a common ratio
between each term.
-/
namespace Real
/--
A `0th`-indexed geometric sequence.
-/
structure Geometric where
a₀ : Real
r : Real
namespace Geometric
/--
Returns the value of the `n`th term of a geometric sequence.
This function calculates the value of this term directly. Keep in mind the
sequence is `0`th-indexed.
-/
def termClosed (seq : Geometric) (n : Nat) : Real :=
seq.a₀ * seq.r ^ n
/--
Returns the value of the `n`th term of a geometric sequence.
This function calculates the value of this term recursively. Keep in mind the
sequence is `0`th-indexed.
-/
def termRecursive : Geometric → Nat → Real
| seq, 0 => seq.a₀
| seq, (n + 1) => seq.r * (seq.termRecursive n)
/--
The recursive and closed term definitions of a geometric sequence agree with
one another.
-/
theorem term_recursive_closed (seq : Geometric) (n : Nat)
: seq.termRecursive n = seq.termClosed n := by
induction n with
| zero => unfold termClosed termRecursive; norm_num
| succ n ih => calc
seq.termRecursive (n + 1)
_ = seq.r * (seq.termRecursive n) := rfl
_ = seq.r * (seq.termClosed n) := by rw [ih]
_ = seq.r * (seq.a₀ * seq.r ^ n) := rfl
_ = seq.a₀ * seq.r ^ (n + 1) := by ring
_ = seq.termClosed (n + 1) := rfl
/--
The summation of the first `n + 1` terms of a geometric sequence.
This function calculates the sum directly.
-/
noncomputable def sum_closed_ratio_neq_one (seq : Geometric) (n : Nat)
: seq.r ≠ 1 → Real :=
fun _ => (seq.a₀ * (1 - seq.r ^ (n + 1))) / (1 - seq.r)
/--
The summation of the first `n + 1` terms of a geometric sequence.
This function calculates the sum recursively.
-/
def sum_recursive : Geometric → Nat → Real
| seq, 0 => seq.a₀
| seq, (n + 1) => seq.termClosed (n + 1) + seq.sum_recursive n
/--
The recursive and closed definitions of the sum of a geometric sequence agree
with one another.
-/
theorem sum_recursive_closed (seq : Geometric) (n : Nat) (p : seq.r ≠ 1)
: sum_recursive seq n = sum_closed_ratio_neq_one seq n p := by
have h : 1 - seq.r ≠ 0 := by
intro h
rw [sub_eq_iff_eq_add, zero_add] at h
exact False.elim (p (Eq.symm h))
induction n with
| zero =>
unfold sum_recursive sum_closed_ratio_neq_one
simp
rw [mul_div_assoc, div_self h, mul_one]
| succ n ih =>
calc
sum_recursive seq (n + 1)
_ = seq.termClosed (n + 1) + seq.sum_recursive n := rfl
_ = seq.termClosed (n + 1) + sum_closed_ratio_neq_one seq n p := by rw [ih]
_ = seq.a₀ * seq.r ^ (n + 1) + (seq.a₀ * (1 - seq.r ^ (n + 1))) / (1 - seq.r) := rfl
_ = seq.a₀ * seq.r ^ (n + 1) * (1 - seq.r) / (1 - seq.r) + (seq.a₀ * (1 - seq.r ^ (n + 1))) / (1 - seq.r) := by rw [mul_div_cancel _ h]
_ = (seq.a₀ * (1 - seq.r ^ (n + 1 + 1))) / (1 - seq.r) := by ring_nf
_ = sum_closed_ratio_neq_one seq (n + 1) p := rfl
end Real.Geometric