Quiz — CoursSeriesEnt.pdf

Question 1 sur 10 20:00

[{"id":85280,"question":"Quelle est la nature de la série de terme général u_n = 1\/n² ?","option_a":"Convergente (série de Riemann)","option_b":"Divergente","option_c":"Oscillante","option_d":"Convergente vers π²\/6","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série de Riemann Σ1\/n^p converge si et seulement si p \u003E 1. Ici, p=2 \u003E 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\": \"Convergente (série de Riemann)\", \"b\": \"Divergente\", \"c\": \"Oscill","_debug_options_count":4},{"id":85281,"question":"Le critère de d'Alembert permet de conclure à la convergence d'une série si la limite du rapport |u_{n+1}\/u_n| est strictement inférieure à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le critère de d'Alembert stipule que si lim |u_{n+1}\/u_n| = L \u003C 1, alors la série converge absolument. La réponse est donc 'Vrai'.","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":85282,"question":"Quelle est la somme de la série entière Σ (x^n)\/n pour |x| \u003C 1 ?","option_a":"-ln(1-x)","option_b":"ln(1+x)","option_c":"e^x","option_d":"arctan(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La série entière Σ (x^n)\/n est le développement en série de Taylor de la fonction ln(1+x) 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\": \"-ln(1-x)\", \"b\": \"ln(1+x)\", \"c\": \"e^x\", \"d\": \"arctan(x)\"}}","_debug_options_count":4},{"id":85283,"question":"Une série alternée Σ (-1)^n * u_n converge si la suite (u_n) est décroissante et tend vers 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le critère de Leibniz pour les séries alternées : si (u_n) est décroissante et tend vers 0, alors 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":85284,"question":"Quel est le rayon de convergence de la série entière Σ n² * x^n ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"2","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le rayon de convergence R est donné par 1\/R = lim |a_{n+1}\/a_n|. Ici, a_n = n², donc lim |(n+1)²\/n²| = 1, ce qui donne R = ∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"∞\", \"d\": \"2\"}}","_debug_options_count":4},{"id":85285,"question":"La série Σ 1\/n! converge-t-elle ? Si oui, vers quelle valeur ?","option_a":"Non, elle diverge","option_b":"Oui, vers e","option_c":"Oui, vers π","option_d":"Oui, vers 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La série Σ 1\/n! converge vers e (constante d'Euler), car c'est le développement en série de e^x évalué en x=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\": \"Non, elle diverge\", \"b\": \"Oui, vers e\", \"c\": \"Oui, vers π\", \"d\":","_debug_options_count":4},{"id":85286,"question":"Le développement limité de ln(1+x) à l'ordre 3 est :","option_a":"x - x²\/2 + x³\/3","option_b":"x - x²\/2 + x³\/3 - x⁴\/4","option_c":"x - x²\/2","option_d":"x + x²\/2 + x³\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le développement limité de ln(1+x) à l'ordre 3 est x - x²\/2 + x³\/3 + o(x³).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x - x²\/2 + x³\/3\", \"b\": \"x - x²\/2 + x³\/3 - x⁴\/4\", \"c\": \"x - ","_debug_options_count":4},{"id":85287,"question":"Une série à termes positifs converge si et seulement si sa suite des sommes partielles est majorée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une série à termes positifs, la convergence équivaut à la majoration de la suite des sommes partielles (théorème de convergence monotone).","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":85288,"question":"Quelle est la nature de la série Σ (-1)^n \/ √n ?","option_a":"Convergente (critère de Leibniz)","option_b":"Divergente","option_c":"Oscillante","option_d":"Convergente (critère de Cauchy)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série Σ (-1)^n \/ √n est une série alternée qui vérifie le critère de Leibniz : (1\/√n) est décroissante et tend vers 0. Elle converge donc.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Convergente (critère de Leibniz)\", \"b\": \"Divergente\", \"c\": \"Osci","_debug_options_count":4},{"id":85289,"question":"Le produit de Cauchy de deux séries convergentes est toujours convergent.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le produit de Cauchy de deux séries convergentes n'est pas nécessairement convergent (contre-exemple : Σ (-1)^n \/ √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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]