Quiz — Magazine 4 Suites réelles.pdf

Question 1 sur 10 20:00

[{"id":99895,"question":"Quelle est la limite de la suite définie par u_n = 3 + 1\/n lorsque n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. +∞","option_d":"D. N'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite u_n = 3 + 1\/n tend vers 3 car 1\/n tend vers 0 lorsque n tend vers l'infini.","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. 3\", \"c\": \"C. +∞\", \"d\": \"D. N'existe pas\"}}","_debug_options_count":4},{"id":99896,"question":"Une suite strictement croissante est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Une suite strictement croissante peut diverger vers +∞ (exemple : u_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},{"id":99897,"question":"Quelle propriété doit vérifier une suite pour être convergente ?","option_a":"A. Être bornée","option_b":"B. Être croissante","option_c":"C. Être décroissante","option_d":"D. Avoir une limite finie","option_e":"","option_f":"","bonne_reponse":"d","explication":"Une suite convergente doit avoir une limite finie, mais elle n'est pas nécessairement bornée (exemple : u_n = (-1)^n \/ n).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A. Être bornée\", \"b\": \"B. Être croissante\", \"c\": \"C. Être dé","_debug_options_count":4},{"id":99898,"question":"La suite définie par u_0 = 1 et u_{n+1} = 2u_n admet pour limite :","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"Cette suite est géométrique de raison 2, donc elle diverge vers +∞.","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. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":99899,"question":"Toute suite bornée est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Une suite bornée peut ne pas avoir de limite (exemple : u_n = (-1)^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},{"id":99900,"question":"Quelle est la limite de la suite u_n = n² \/ (n² + 1) ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n², on obtient u_n = 1 \/ (1 + 1\/n²), qui tend vers 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":99901,"question":"Une suite définie par u_{n+1} = f(u_n) converge si :","option_a":"A. f est continue","option_b":"B. f est croissante","option_c":"C. f est contractante","option_d":"D. f est décroissante","option_e":"","option_f":"","bonne_reponse":"c","explication":"Une suite définie par u_{n+1} = f(u_n) converge si f est contractante (théorème du point fixe).","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. f est continue\", \"b\": \"B. f est croissante\", \"c\": \"C. f est co","_debug_options_count":4},{"id":99902,"question":"La suite u_n = (-1)^n est bornée mais non convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Cette suite est bornée (entre -1 et 1) mais ne converge pas car elle oscille indéfiniment.","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":99903,"question":"Quelle est la limite de la suite u_n = (2n + 1) \/ (n - 3) ?","option_a":"A. 0","option_b":"B. 2","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n, on obtient u_n = (2 + 1\/n) \/ (1 - 3\/n), qui tend vers 2.","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. 2\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":99904,"question":"Une suite convergente est toujours monotone.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Une suite convergente n'est pas nécessairement monotone (exemple : u_n = sin(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}]