mathlib3 documentation

analysis.sum_integral_comparisons

Comparing sums and integrals #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

Summary #

It is often the case that error terms in analysis can be computed by comparing an infinite sum to the improper integral of an antitone function. This file will eventually enable that.

At the moment it contains four lemmas in this direction: antitone_on.integral_le_sum, antitone_on.sum_le_integral and versions for monotone functions, which can all be paired with a filter.tendsto to estimate some errors.

TODO: Add more lemmas to the API to directly address limiting issues

Main Results #

Tags #

analysis, comparison, asymptotics

theorem antitone_on.integral_le_sum {x₀ : } {a : } {f : } (hf : antitone_on f (set.Icc x₀ (x₀ + a))) :
(x : ) in x₀..x₀ + a, f x (finset.range a).sum (λ (i : ), f (x₀ + i))
theorem antitone_on.integral_le_sum_Ico {a b : } {f : } (hab : a b) (hf : antitone_on f (set.Icc a b)) :
(x : ) in a..b, f x (finset.Ico a b).sum (λ (x : ), f x)
theorem antitone_on.sum_le_integral {x₀ : } {a : } {f : } (hf : antitone_on f (set.Icc x₀ (x₀ + a))) :
(finset.range a).sum (λ (i : ), f (x₀ + (i + 1))) (x : ) in x₀..x₀ + a, f x
theorem antitone_on.sum_le_integral_Ico {a b : } {f : } (hab : a b) (hf : antitone_on f (set.Icc a b)) :
(finset.Ico a b).sum (λ (i : ), f (i + 1)) (x : ) in a..b, f x
theorem monotone_on.sum_le_integral {x₀ : } {a : } {f : } (hf : monotone_on f (set.Icc x₀ (x₀ + a))) :
(finset.range a).sum (λ (i : ), f (x₀ + i)) (x : ) in x₀..x₀ + a, f x
theorem monotone_on.sum_le_integral_Ico {a b : } {f : } (hab : a b) (hf : monotone_on f (set.Icc a b)) :
(finset.Ico a b).sum (λ (x : ), f x) (x : ) in a..b, f x
theorem monotone_on.integral_le_sum {x₀ : } {a : } {f : } (hf : monotone_on f (set.Icc x₀ (x₀ + a))) :
(x : ) in x₀..x₀ + a, f x (finset.range a).sum (λ (i : ), f (x₀ + (i + 1)))
theorem monotone_on.integral_le_sum_Ico {a b : } {f : } (hab : a b) (hf : monotone_on f (set.Icc a b)) :
(x : ) in a..b, f x (finset.Ico a b).sum (λ (i : ), f (i + 1))