proposition: Lemma sub_eseries m n : eseries m \is a fin_num -> eseries n \is a fin_num -> eseries n - eseries m = if (m <= n)%N then \sum_(m <= k < n) u_ k else - \sum_(n <= k < m) u_ k.
proof: Proof. move=> ? ?; have [mn|/ltnW mn] := leqP m n; rewrite -sub_eseries_geq//. by rewrite fin_num_oppeD ?fin_numN// oppeK addeC. Qed.
imports: mathcomp(all_ssreflect ssralg ssrint ssrnum matrix) mathcomp(interval rat) mathcomp(mathcomp_extra boolp classical_sets functions) mathcomp(set_interval) reals ereal signed topology normedtype landau