Quiz — Concours des Mines 1996 (Séries associées à la fonction zeta de Riemann).pdf

Question 1 sur 10 20:00

[{"id":85665,"question":"Quelle est la valeur de ζ(2) selon la formule d'Euler ?","option_a":"π²\/6","option_b":"π²\/4","option_c":"π\/2","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule d'Euler donne ζ(2) = Σ 1\/n² = π²\/6, un résultat fondamental en mathématiques.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"π²\/6\", \"b\": \"π²\/4\", \"c\": \"π\/2\", \"d\": \"1\"}}","_debug_options_count":4},{"id":85666,"question":"La série Σ (-1)^(n+1)\/n converge-t-elle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la série harmonique alternée, qui converge par le critère de Leibniz (termes décroissants en valeur absolue et tendant vers 0).","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":85667,"question":"Quel critère permet de montrer la convergence de Σ 1\/(n ln n)² ?","option_a":"Critère de Cauchy","option_b":"Critère de Raabe-Duhamel","option_c":"Critère de comparaison série-intégrale","option_d":"Critère d'Abel","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le critère de comparaison série-intégrale est adapté ici, car la fonction f(x) = 1\/(x ln x)² est positive, décroissante et intégrable sur [2, +∞[.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Critère de Cauchy\", \"b\": \"Critère de Raabe-Duhamel\", \"c\": \"Crit","_debug_options_count":4},{"id":85668,"question":"La fonction zêta de Riemann ζ(s) est-elle définie pour s = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"ζ(1) correspond à la série harmonique Σ 1\/n, qui diverge. La fonction zêta est définie par prolongement analytique pour Re(s) \u003E 0, s ≠ 1.","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":85669,"question":"Quelle égalité est vérifiée par la série Σ 1\/n(n+1) ?","option_a":"Σ 1\/n(n+1) = 1","option_b":"Σ 1\/n(n+1) = 1\/2","option_c":"Σ 1\/n(n+1) = 2","option_d":"La série diverge","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par télescopage, Σ 1\/n(n+1) = Σ (1\/n - 1\/(n+1)) = 1 (série convergente).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Σ 1\/n(n+1) = 1\", \"b\": \"Σ 1\/n(n+1) = 1\/2\", \"c\": \"Σ 1\/n(n+1) = 2","_debug_options_count":4},{"id":85670,"question":"Le critère de d'Alembert s'applique-t-il à la série Σ n!\/n^n ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, car la limite de (u_{n+1}\/u_n) = (n+1)\/n * 1\/(1+1\/n)^n tend vers 1\/e \u003C 1, donc la série 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":85671,"question":"Quelle est la nature de la série Σ (-1)^n \/ ln(n) pour n ≥ 2 ?","option_a":"Convergente absolument","option_b":"Convergente simplement","option_c":"Divergente","option_d":"Indéterminée","option_e":"","option_f":"","bonne_reponse":"b","explication":"La série converge simplement par le critère de Leibniz (termes décroissants en valeur absolue et tendant vers 0), mais pas absolument (série harmonique alternée des ln(n)).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Convergente absolument\", \"b\": \"Convergente simplement\", \"c\": \"Div","_debug_options_count":4},{"id":85672,"question":"La fonction ζ(s) peut-elle être prolongée en une fonction méromorphe sur ℂ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, par prolongement analytique, ζ(s) est méromorphe sur ℂ avec un unique pôle simple en s = 1 de résidu 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":85673,"question":"Quelle inégalité est vérifiée par ζ(s) pour s \u003E 1 ?","option_a":"ζ(s) \u003E 1","option_b":"ζ(s) \u003C 1","option_c":"ζ(s) = 1","option_d":"ζ(s) est indéfinie","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour s \u003E 1, ζ(s) = 1 + 1\/2^s + 1\/3^s + ... \u003E 1, car tous les termes sont strictement positifs.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"ζ(s) \u003E 1\", \"b\": \"ζ(s) \u003C 1\", \"c\": \"ζ(s) = 1\", \"d\": \"ζ(s) est i","_debug_options_count":4},{"id":85674,"question":"Quel est le rayon de convergence de la série entière Σ ζ(n) x^n ?","option_a":"0","option_b":"1","option_c":"+∞","option_d":"Indéterminé","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le rayon de convergence est R = 1, car ζ(n) tend vers 1 quand n → ∞, et la série géométrique Σ x^n converge pour |x| \u003C 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"+∞\", \"d\": \"Indéterminé\"}}","_debug_options_count":4}]