Definition is_pseries (a : nat -> V) (x:K) (l : V) :=
  is_series (fun k => scal (pow_n x k) (a k)) l.Definition ex_pseries (a : nat -> V) (x : K) :=
  ex_series (fun k => scal (pow_n x k) (a k)).Definition PSeries (a : nat -> R) (x : R) : R :=
  Series (fun k => a k * x ^ k).

(* Specifications: *)
Lemma ex_pseries_dec {V : NormedModule R_AbsRing} (a : nat -> R) (x : R) : {ex_pseries a x} + {~ ex_pseries a x}.
Lemma is_pseries_R (a : nat -> R) (x l : R) : is_pseries a x l <-> is_series (fun n : nat => a n * x ^ n) l.
Lemma ex_pseries_R (a : nat -> R) (x : R) : ex_pseries a x <-> ex_series (fun n : nat => a n * x ^ n).
Lemma PSeries_eq (a : nat -> R) (x : R) : PSeries a x = Series (fun k => scal (pow_n x k) (a k)).
Lemma PSeries_1 (a : nat -> R) : PSeries a 1 = Series a.
Lemma ex_pseries_1 (a : nat -> R) : ex_pseries a 1 <-> ex_series a.
Lemma is_pseries_unique (a : nat -> R) (x l : R) : is_pseries a x l -> PSeries a x = l.
Lemma PSeries_correct (a : nat -> R) (x : R) : ex_pseries a x -> is_pseries a x (PSeries a x).
Lemma is_pseries_Reals (a : nat -> R) (x l : R) : is_pseries a x l <-> Pser a x l.
Lemma is_pseries_ext (a b : nat -> V) (x : K) (l:V) : (forall n, a n = b n) -> (is_pseries a x l) -> is_pseries b x l.
Lemma ex_pseries_ext (a b : nat -> V) (x : K) : (forall n, a n = b n) -> ex_pseries a x -> ex_pseries b x.
Lemma PSeries_ext (a b : nat -> R) (x : R) : (forall n, a n = b n) -> PSeries a x = PSeries b x.
Lemma is_pseries_0 (a : nat -> V) : is_pseries a zero (a O).
Lemma ex_pseries_0 (a : nat -> V) : ex_pseries a zero.
Lemma PSeries_0 (a : nat -> R) : PSeries a 0 = a O.