Section Properties3.
Context {K : AbsRing} {V : NormedModule K}.
exact is_series_scal.
Qed.
exact ex_series_scal.
Qed.
End Properties3.
(** * D'Alembert criterion *)

Lemma partial_summation_R (a b : nat -> R) :
(exists M, forall n, norm (sum_n b n) <= M)
-> filterlim a eventually (locally 0)
-> ex_series (fun n => norm (minus (a (S n)) (a n)))
-> ex_series (fun n => scal (a n) (b n)).