Definition is_series (a : nat -> V) (l : V) :=
   filterlim (sum_n a) (eventually) (locally l).Definition ex_series (a : nat -> V) :=
   exists l : V, is_series a l.Definition Cauchy_series (a : nat -> V) :=
  forall eps : posreal,
    exists N : nat, forall n m : nat,
      (N <= n)%nat -> (N <= m)%nat ->
      norm (sum_n_m a n m) < eps.Definition Series (a : nat -> R) : R :=
   real (Lim_seq (sum_n a)).

Lemma is_series_mult_pos (a b : nat -> R) (la lb : R) : is_series a la -> is_series b lb -> (forall n, 0 <= a n) -> (forall n, 0 <= b n) -> is_series (fun n => sum_f_R0 (fun k => a k * b (n - k)%nat) n) (la * lb).