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 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)).