proposition: Fact is_cvg_pseries_inside f (x z : R) : cvgn (pseries f x) -> `|z| < `|x| -> cvgn (pseries f z).
proof: Proof. move=> Cx zLx. apply: normed_cvg; rewrite /normed_series_of /=. rewrite (_ : (fun _ => _) = (fun i => `|f i| * `|z| ^+ i)); last first. by apply/funext => i; rewrite normrM normrX. by apply: is_cvg_pseries_inside_norm Cx _; rewrite normr_id. Qed.
imports: mathcomp(all_ssreflect ssralg ssrint ssrnum matrix) mathcomp(interval rat) mathcomp(boolp classical_sets functions) mathcomp(mathcomp_extra) reals ereal signed topology normedtype landau sequences derive realfun itv convex