proposition: Lemma diffs_cos : pseries_diffs (fun n => (~~(odd n))%:R * (-1) ^+ n./2 * (n`!%:R)^-1) = (fun n => - ((odd n)%:R * (-1) ^+ n.-1./2 * (n`!%:R)^-1): R).
proof: Proof. apply/funext => [] [|i] /=. by rewrite /pseries_diffs /= !mul0r mulr0 oppr0. rewrite /pseries_diffs /= negbK exprS mulN1r !(mulNr, mulrN). rewrite factS natrM invfM. by rewrite [_.+1%:R * _]mulrC -!mulrA [_.+1%:R^-1 * _]mulrC mulfK. Qed.
imports: mathcomp(all_ssreflect ssralg ssrint ssrnum matrix) mathcomp(interval rat) mathcomp(mathcomp_extra boolp classical_sets functions) reals ereal nsatz_realtype signed topology normedtype landau sequences derive realfun exp