Lemma derivable_pt_lim_fs : forall (An:nat -> R) (x:R) (N:nat), (0 < N)%nat -> derivable_pt_lim (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N) x (sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred N)).