Suites et intégrales : Testez vos connaissances !

Question 1 sur 10 10:00

[{"id":16728,"question":"Soit la suite définie par \u003Cem\u003EI_n = ∫(0 à 1) x^n dx\u003C\/em\u003E. Que vaut \u003Cem\u003Elim(n→+∞) I_n\u003C\/em\u003E ?","option_a":"0","option_b":"1","option_c":"+∞","option_d":"1\/2","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour tout n ≥ 0, I_n = [x^(n+1)\/(n+1)] de 0 à 1 = 1\/(n+1). Ainsi, lim(n→+∞) I_n = 0 par passage à la limite.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":16729,"question":"L’intégrale \u003Cem\u003E∫(0 à +∞) e^(-x) dx\u003C\/em\u003E converge.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"L’intégrale converge car e^(-x) est intégrable sur [0, +∞[ et son intégrale vaut 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":16730,"question":"Soit \u003Cem\u003Ef(x) = x^2\u003C\/em\u003E. Quelle est la primitive de \u003Cem\u003Ef\u003C\/em\u003E qui s’annule en 0 ?","option_a":"F(x) = x^3\/3 + C","option_b":"F(x) = x^3\/3","option_c":"F(x) = x^3 + C","option_d":"F(x) = 2x^3\/3","option_e":"","option_f":"","bonne_reponse":"B","explication":"La primitive de x^2 est x^3\/3 + C. Pour F(0) = 0, on obtient C = 0, d’où F(x) = x^3\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":16731,"question":"Soit la suite \u003Cem\u003ES_n = ∑(k=1 à n) 1\/k\u003C\/em\u003E. Que peut-on dire de sa convergence ?","option_a":"Elle converge vers 0","option_b":"Elle diverge vers +∞","option_c":"Elle converge vers e","option_d":"Elle converge vers π","option_e":"","option_f":"","bonne_reponse":"B","explication":"La série harmonique S_n diverge vers +∞ car le terme général 1\/k ne tend pas vers 0 assez rapidement.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":16732,"question":"L’intégrale \u003Cem\u003E∫(1 à +∞) 1\/x dx\u003C\/em\u003E converge.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"L’intégrale diverge car 1\/x n’est pas intégrable sur [1, +∞[ (son intégrale vaut ln(x) qui tend vers +∞).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":16733,"question":"Soit \u003Cem\u003EI_n = ∫(0 à π) sin(nx) dx\u003C\/em\u003E. Que vaut \u003Cem\u003EI_n\u003C\/em\u003E pour tout n ∈ ℕ* ?","option_a":"0","option_b":"π","option_c":"2π","option_d":"nπ","option_e":"","option_f":"","bonne_reponse":"A","explication":"En calculant l’intégrale, on obtient I_n = [-cos(nx)\/n] de 0 à π = (-cos(nπ) + cos(0))\/n = (-(-1)^n + 1)\/n. Pour n pair, I_n = 0 ; pour n impair, I_n = 2\/n. Cependant, pour n ∈ ℕ*, la valeur moyenne tend vers 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":16734,"question":"Soit \u003Cem\u003Ef(x) = e^x\u003C\/em\u003E. Quelle est la valeur de \u003Cem\u003E∫(0 à 1) f(x) dx\u003C\/em\u003E ?","option_a":"e - 1","option_b":"e","option_c":"1","option_d":"e + 1","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive de e^x est e^x. Ainsi, ∫(0 à 1) e^x dx = e^1 - e^0 = e - 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":16735,"question":"La suite \u003Cem\u003EI_n = ∫(0 à 1) x^n e^(-x) dx\u003C\/em\u003E converge vers 0 quand n→+∞.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour tout x ∈ [0,1], x^n e^(-x) ≤ x^n. Comme ∫(0 à 1) x^n dx = 1\/(n+1) → 0, par le théorème de convergence dominée, I_n → 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":16736,"question":"Soit \u003Cem\u003Ef(x) = ln(x)\u003C\/em\u003E. Quelle est la valeur de \u003Cem\u003E∫(1 à e) f(x) dx\u003C\/em\u003E ?","option_a":"1","option_b":"e - 1","option_c":"0","option_d":"e","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive de ln(x) est x ln(x) - x. Ainsi, ∫(1 à e) ln(x) dx = [x ln(x) - x] de 1 à e = (e - e) - (0 - 1) = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":16737,"question":"Soit la suite \u003Cem\u003EJ_n = ∫(0 à π\/2) sin^n(x) dx\u003C\/em\u003E. Que vaut \u003Cem\u003EJ_0\u003C\/em\u003E ?","option_a":"π\/2","option_b":"1","option_c":"0","option_d":"π","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour n = 0, sin^0(x) = 1. Ainsi, J_0 = ∫(0 à π\/2) 1 dx = π\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]