Documentation

Mathlib.Algebra.BigOperators.Module

Summation by parts #

theorem Finset.sum_Ico_by_parts {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (f : R) (g : M) {m : } {n : } (hmn : m < n) :
iFinset.Ico m n, f i g i = f (n - 1) iFinset.range n, g i - f m iFinset.range m, g i - iFinset.Ico m (n - 1), (f (i + 1) - f i) iFinset.range (i + 1), g i

Summation by parts, also known as Abel's lemma or an Abel transformation

theorem Finset.sum_range_by_parts {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (f : R) (g : M) (n : ) :
iFinset.range n, f i g i = f (n - 1) iFinset.range n, g i - iFinset.range (n - 1), (f (i + 1) - f i) iFinset.range (i + 1), g i

Summation by parts for ranges