Section Definitions.
(** * Definition *)
Context {K : AbsRing} {V : NormedModule K}.
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)).
End Definitions.
Definition PSeries (a : nat -> R) (x : R) : R :=
Series (fun k => a k * x ^ k).

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.