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 Series_incr_n (a : nat -> R) (n : nat) : (0 < n)%nat -> ex_series a -> Series a = sum_f_R0 a (pred n) + Series (fun k => a (n + k)%nat).