Partial sums of geometric series in an ordered field #
This file upper- and lower-bounds the values of the geometric series $\sum_{i=0}^{n-1} x^i$ and $\sum_{i=0}^{n-1} x^i y^{n-1-i}$ and variants thereof.
theorem
geom₂_sum_of_gt
{K : Type u_1}
[Semifield K]
[LinearOrder K]
[IsStrictOrderedRing K]
[CanonicallyOrderedAdd K]
[Sub K]
[OrderedSub K]
{x y : K}
(h : y < x)
(n : ℕ)
:
theorem
geom₂_sum_of_lt
{K : Type u_1}
[Semifield K]
[LinearOrder K]
[IsStrictOrderedRing K]
[CanonicallyOrderedAdd K]
[Sub K]
[OrderedSub K]
{x y : K}
(h : x < y)
(n : ℕ)
:
theorem
geom_sum_of_one_lt
{K : Type u_1}
[Semifield K]
[LinearOrder K]
[IsStrictOrderedRing K]
[CanonicallyOrderedAdd K]
[Sub K]
[OrderedSub K]
{x : K}
(h : 1 < x)
(n : ℕ)
:
theorem
geom_sum_of_lt_one
{K : Type u_1}
[Semifield K]
[LinearOrder K]
[IsStrictOrderedRing K]
[CanonicallyOrderedAdd K]
[Sub K]
[OrderedSub K]
{x : K}
(h : x < 1)
(n : ℕ)
:
theorem
geom_sum_lt
{K : Type u_1}
[Semifield K]
[LinearOrder K]
[IsStrictOrderedRing K]
[CanonicallyOrderedAdd K]
[Sub K]
[OrderedSub K]
{x : K}
(h0 : x ≠ 0)
(h1 : x < 1)
(n : ℕ)
:
theorem
geom_sum_Ico_le_of_lt_one
{K : Type u_1}
[Field K]
[LinearOrder K]
[IsStrictOrderedRing K]
{x : K}
{m n : ℕ}
(hx : 0 ≤ x)
(h'x : x < 1)
: