(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
Local Open Scope R_scope.

Lemma derivable_pt_lim_finite_sum :
forall (An:nat -> R) (x:R) (N:nat),
derivable_pt_lim (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N) x
match N with
| O => 0
| _ => sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred N)
end.