(* More efficient way to compute approximations of PI. *)

Definition PI_2_3_7_tg n :=
  2 * Ratan_seq (/3) n + Ratan_seq (/7) n.

Lemma PI_2_3_7_ineq : forall N : nat, sum_f_R0 (tg_alt PI_2_3_7_tg) (S (2 * N)) <= PI / 4 <= sum_f_R0 (tg_alt PI_2_3_7_tg) (2 * N).