Théorèmes de comparaison

FondamentalThéorème

Si  ∃ n 0 ∈ ℕ , ∀ n ∈ ℕ , n ≥ n 0 ⇒ u n ≤ v n exists n_0 in setN, forall n in setN, ` n >= n_0 `drarrow` u_n <= v_n et lim n → + ∞ u n = + ∞ lim from {n toward +infinity} u_n = + infinity alors lim n → + ∞ v n = + ∞ lim from {n toward +infinity} v_n = + infinity

FondamentalThéorème

Soient u n u_n , v n v_n et w n w_n trois suites telles que :

Si ∃ n 0 ∈ ℕ , ∀ n ∈ ℕ , n ≥ n 0 ⇒ u n ≤ v n ≤ w n exists n_0 in setN, forall n in setN, ` n >= n_0 `drarrow` u_n <= v_n <= w_n et lim n → + ∞ u n = lim n → + ∞ w n = l lim from {n toward +infinity} u_n = lim from {n toward +infinity} w_n = l alors ( v n ) (v_n) converge et lim n → + ∞ v n = l lim from {n toward +infinity} v_n = l

Exemple

Soit la suite ( u n ) (u_n) définie par : u n = sin n n u_n = {sin n} over n

On a − 1 n ≤ sin n n ≤ 1 n -{1 over n} <= {sin n} over n <= {1 over n} donc lim n → + ∞ sin n n = 0 lim from {n toward +infinity} {{sin n} over n} = 0