Quiz — test1serie numerique.pdf

Question 1 sur 10 20:00

[{"id":54567,"question":"Quelle série numérique est convergente selon le critère de d'Alembert ?","option_a":"u_n = 1\/n","option_b":"u_n = n\/2^n","option_c":"u_n = (-1)^n \/ n","option_d":"u_n = n^2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour u_n = n\/2^n, on applique le critère de d'Alembert : lim |u_{n+1}\/u_n| = lim (n+1)\/2n = 1\/2 \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\": \"b\", \"options\": {\"a\": \"u_n = 1\/n\", \"b\": \"u_n = n\/2^n\", \"c\": \"u_n = (-1)^n \/ n\", \"d\": \"u_","_debug_options_count":4},{"id":54568,"question":"La série harmonique Σ 1\/n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La série harmonique Σ 1\/n est un exemple classique de série divergente, comme le montre le critère de comparaison avec l'intégrale de 1\/x.","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":54569,"question":"Quelle est la somme de la série géométrique Σ (1\/2)^n pour n allant de 0 à l'infini ?","option_a":"0","option_b":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La somme d'une série géométrique Σ r^n (|r| \u003C 1) est 1\/(1-r). Ici, r = 1\/2, donc la somme vaut 1\/(1-1\/2) = 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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":54570,"question":"Si lim (u_{n+1}\/u_n) = 1, que peut-on conclure sur la série Σ u_n ?","option_a":"La série converge","option_b":"La série diverge","option_c":"Le critère de d'Alembert ne permet pas de conclure","option_d":"La série est géométrique","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le critère de d'Alembert est indécis lorsque la limite vaut 1. D'autres critères (Cauchy, comparaison) doivent être utilisés.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"La série converge\", \"b\": \"La série diverge\", \"c\": \"Le critère ","_debug_options_count":4},{"id":54571,"question":"La série Σ (-1)^n \/ n^2 est-elle absolument convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série Σ |(-1)^n \/ n^2| = Σ 1\/n^2 est convergente (série de Riemann avec α = 2 \u003E 1), donc la série est absolument 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":54572,"question":"Quel critère utilise-t-on pour étudier la convergence de la série Σ 1\/n^α ?","option_a":"Critère de d'Alembert","option_b":"Critère de Cauchy","option_c":"Critère de comparaison","option_d":"Critère de Riemann","option_e":"","option_f":"","bonne_reponse":"d","explication":"Le critère de Riemann (ou critère de l'intégrale) permet de déterminer la convergence de la série Σ 1\/n^α en fonction de la valeur de α.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"Critère de d'Alembert\", \"b\": \"Critère de Cauchy\", \"c\": \"Critèr","_debug_options_count":4},{"id":54573,"question":"Si une série Σ u_n converge, alors la suite (u_n) tend vers 0. Cette affirmation est-elle vraie ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une condition nécessaire (mais non suffisante) pour la convergence d'une série : si Σ u_n converge, alors lim u_n = 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":54574,"question":"Quelle est la nature de la série Σ n \/ (n^2 + 1) ?","option_a":"Convergente","option_b":"Divergente","option_c":"Semi-convergente","option_d":"Géométrique","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant le critère de comparaison avec la série harmonique Σ 1\/n, on montre que Σ n\/(n^2+1) diverge.","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\", \"b\": \"Divergente\", \"c\": \"Semi-convergente\", \"d\": \"G","_debug_options_count":4},{"id":54575,"question":"Le critère de Cauchy (ou critère de la racine) s'applique-t-il à la série Σ (1 + 1\/n)^n ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le critère de Cauchy ne s'applique pas ici car le terme général (1 + 1\/n)^n ne tend pas vers 0 (il tend vers e). La série diverge donc.","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":54576,"question":"Quelle série est semi-convergente parmi les suivantes ?","option_a":"Σ 1\/n^2","option_b":"Σ (-1)^n \/ n","option_c":"Σ 1\/2^n","option_d":"Σ n\/2^n","option_e":"","option_f":"","bonne_reponse":"b","explication":"La série Σ (-1)^n \/ n est convergente (critère des séries alternées) mais pas absolument convergente (car Σ 1\/n diverge). Elle est donc semi-convergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Σ 1\/n^2\", \"b\": \"Σ (-1)^n \/ n\", \"c\": \"Σ 1\/2^n\", \"d\": \"Σ n\/2^n\"","_debug_options_count":4}]