Quiz — TD_Series_numeriques (1).pdf

Question 1 sur 10 20:00

[{"id":79031,"question":"Quelle série est convergente parmi les suivantes ?","option_a":"A. Σ (1\/n)","option_b":"B. Σ (1\/n²)","option_c":"C. Σ (n)","option_d":"D. Σ (1\/√n)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La série Σ (1\/n²) est une série de Riemann convergente (p=2\u003E1). Les autres séries sont divergentes (série harmonique, série linéaire, série p avec p=1\/2\u003C1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. Σ (1\/n)\", \"b\": \"B. Σ (1\/n²)\", \"c\": \"C. Σ (n)\", \"d\": \"D. Σ","_debug_options_count":4},{"id":79032,"question":"Le critère de d'Alembert s'applique-t-il à la série Σ (n! \/ 2ⁿ) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. La série est à termes positifs et le rapport (n+1)!\/2^(n+1) \/ (n!\/2ⁿ) = (n+1)\/2 tend vers l'infini, donc la série diverge par le critère de d'Alembert.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":79033,"question":"Pour étudier la convergence de Σ (sin(n)\/n²), quel critère utilisez-vous ?","option_a":"A. Critère de comparaison","option_b":"B. Critère de Cauchy","option_c":"C. Critère de Leibniz","option_d":"D. Critère de d'Alembert","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise le critère de comparaison car |sin(n)\/n²| ≤ 1\/n², et Σ (1\/n²) est convergente (série de Riemann).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. Critère de comparaison\", \"b\": \"B. Critère de Cauchy\", \"c\": \"","_debug_options_count":4},{"id":79034,"question":"La série Σ ((-1)ⁿ \/ n) est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. C'est une série alternée qui vérifie le critère de Leibniz (termes décroissants en valeur absolue et tendant vers 0), donc elle converge.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":79035,"question":"Si lim (u_{n+1}\/u_n) = 1, que peut-on conclure sur la série Σ u_n ?","option_a":"A. La série converge","option_b":"B. La série diverge","option_c":"C. Le critère de d'Alembert ne permet pas de conclure","option_d":"D. La série est alternée","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le critère de d'Alembert ne permet pas de conclure si la limite vaut 1. Il faut utiliser un autre critère (comparaison, intégrale, etc.).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. La série converge\", \"b\": \"B. La série diverge\", \"c\": \"C. Le ","_debug_options_count":4},{"id":79036,"question":"La série Σ (1\/(n ln n)) converge-t-elle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. C'est une série de Bertrand divergente (p=1 et q=1). On peut le montrer par comparaison avec une intégrale impropre divergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":79037,"question":"Quelle est la nature de la série Σ (n² \/ 3ⁿ) ?","option_a":"A. Convergente","option_b":"B. Divergente","option_c":"C. Semi-convergente","option_d":"D. Oscillante","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série est convergente. On peut appliquer le critère de d'Alembert : lim (u_{n+1}\/u_n) = lim ((n+1)² \/ 3^(n+1)) * (3ⁿ \/ n²) = 1\/3 \u003C 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. Convergente\", \"b\": \"B. Divergente\", \"c\": \"C. Semi-convergente\"","_debug_options_count":4},{"id":79038,"question":"Le critère de Cauchy (racine n-ième) s'applique-t-il à toute série ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Le critère de Cauchy ne s'applique qu'aux séries à termes positifs. Pour les séries à termes quelconques, il faut utiliser d'autres méthodes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":79039,"question":"Pour la série Σ (1\/n - 1\/(n+1)), que vaut la somme partielle S_n ?","option_a":"A. S_n = 1 - 1\/(n+1)","option_b":"B. S_n = 1\/n","option_c":"C. S_n = n","option_d":"D. S_n = 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série est télescopique : S_n = (1 - 1\/2) + (1\/2 - 1\/3) + ... + (1\/n - 1\/(n+1)) = 1 - 1\/(n+1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. S_n = 1 - 1\/(n+1)\", \"b\": \"B. S_n = 1\/n\", \"c\": \"C. S_n = n\", \"d","_debug_options_count":4},{"id":79040,"question":"Si une série Σ u_n converge absolument, que peut-on dire de sa convergence simple ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. La convergence absolue implique la convergence simple (théorème de comparaison des séries).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]