Quiz — 62cc034eeb426_Magazine 11 Suites réelles.pdf

Question 1 sur 10 20:00

[{"id":30280,"question":"Quelle est la limite de la suite définie par u_n = (3n + 1)\/(2n - 5) ?","option_a":"A. 3\/2","option_b":"B. 1","option_c":"C. +∞","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n, on obtient u_n = (3 + 1\/n)\/(2 - 5\/n). Lorsque n tend vers l'infini, 1\/n et 5\/n tendent vers 0, donc la limite est 3\/2.","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. 3\/2\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":30281,"question":"Une suite croissante et majorée est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème de convergence monotone : une suite croissante et majorée converge vers sa borne supérieure.","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":30282,"question":"Soit u_n une suite telle que u_{n+1} = u_n^2 - 2u_n + 2. Que peut-on dire de cette suite si u_0 = 1 ?","option_a":"A. Elle est décroissante","option_b":"B. Elle est constante","option_c":"C. Elle est croissante","option_d":"D. Elle est périodique","option_e":"","option_f":"","bonne_reponse":"b","explication":"Calculons u_1 = 1^2 - 2*1 + 2 = 1. Tous les termes sont égaux à 1, donc la suite est constante.","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. Elle est décroissante\", \"b\": \"B. Elle est constante\", \"c\": \"C","_debug_options_count":4},{"id":30283,"question":"La suite définie par u_n = (-1)^n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette suite oscille entre -1 et 1 et n'a pas de limite. Elle est donc 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":30284,"question":"Quelle est la limite de la suite u_n = n^2 - 5n + 3 ?","option_a":"A. 0","option_b":"B. +∞","option_c":"C. -∞","option_d":"D. 3","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le terme dominant est n^2, donc lorsque n tend vers l'infini, u_n tend vers +∞.","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. 0\", \"b\": \"B. +∞\", \"c\": \"C. -∞\", \"d\": \"D. 3\"}}","_debug_options_count":4},{"id":30285,"question":"Si deux suites u_n et v_n convergent respectivement vers L et M, alors la suite w_n = u_n + v_n converge vers L + M.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale des limites : la limite d'une somme est la somme des limites.","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":30286,"question":"Soit u_n une suite telle que u_{n+1} = (u_n + 2)\/3 avec u_0 = 0. Quelle est la limite de cette suite ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite est contractante et converge vers le point fixe de la fonction f(x) = (x + 2)\/3, soit x = 1. Mais en calculant les premiers termes, on observe une convergence vers 1.","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. 0\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":30287,"question":"Une suite décroissante et minorée est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une conséquence du théorème de convergence monotone : une suite décroissante et minorée converge vers sa borne inférieure.","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":30288,"question":"Quelle est la limite de la suite u_n = (2n + 1)\/(n^2 + 3) ?","option_a":"A. 0","option_b":"B. 2","option_c":"C. +∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le dénominateur est de degré supérieur au numérateur. En divisant par n^2, on obtient u_n = (2\/n + 1\/n^2)\/(1 + 3\/n^2), donc la limite est 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\": \"A. 0\", \"b\": \"B. 2\", \"c\": \"C. +∞\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":30289,"question":"Si u_n est une suite convergente vers L, alors toute sous-suite de u_n converge aussi vers L.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale des suites convergentes : toute sous-suite d'une suite convergente converge vers la même limite.","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}]