End PowerRZ.
#[deprecated(since="8.16",note="Use powerRZ_inv'.")]
Notation powerRZ_inv := powerRZ_inv_depr.
#[deprecated(since="8.16",note="Use powerRZ_neg' and powerRZ_inv'.")]
Notation powerRZ_neg := powerRZ_neg_depr.
#[deprecated(since="8.16",note="Use powerRZ_mult.")]
Notation powerRZ_mult_distr := powerRZ_mult_distr_depr.
Local Infix "^Z" := powerRZ (at level 30, right associativity) : R_scope.
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(* For easy interface          *)
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(* decimal_exp r z is defined as r 10^z *)
Definition decimal_exp (r:R) (z:Z) : R := (r * 10 ^Z z).
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(** * Sum of n first naturals  *)
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Fixpoint sum_nat_f_O (f:nat -> nat) (n:nat) : nat :=
match n with
| O => f 0%nat
| S n' => (sum_nat_f_O f n' + f (S n'))%nat
end.
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Definition sum_nat_f (s n:nat) (f:nat -> nat) : nat :=
sum_nat_f_O (fun x:nat => f (x + s)%nat) (n - s).
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Definition sum_nat_O (n:nat) : nat := sum_nat_f_O (fun x:nat => x) n.
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Definition sum_nat (s n:nat) : nat := sum_nat_f s n (fun x:nat => x).
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(** *          Sum             *)
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Fixpoint sum_f_R0 (f:nat -> R) (N:nat) : R :=
match N with
| O => f 0%nat
| S i => sum_f_R0 f i + f (S i)
end.
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Definition sum_f (s n:nat) (f:nat -> R) : R :=
sum_f_R0 (fun x:nat => f (x + s)%nat) (n - s).

(* Specifications: *)
Lemma GP_finite :
forall (x:R) (n:nat),
sum_f_R0 (fun n:nat => x ^ n) n * (x - 1) = x ^ (n + 1) - 1.
Lemma sum_f_R0_triangle :
forall (x:nat -> R) (n:nat),
Rabs (sum_f_R0 x n) <= sum_f_R0 (fun i:nat => Rabs (x i)) n.