Quiz — Magazine 8 Suites réelles.pdf

Question 1 sur 10 20:00

[{"id":48732,"question":"Quelle est la limite de la suite u_n = (3n + 1)\/(2n - 5) lorsque n tend vers l'infini ?","option_a":"A. 3\/2","option_b":"B. 0","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite d'un quotient de polynômes est égale au rapport des coefficients des termes de plus haut degré, soit 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. 0\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":48733,"question":"Une suite croissante et non majorée est-elle toujours divergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite croissante et non majorée tend vers +∞, donc elle est divergente.","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":48734,"question":"Soit la suite définie par u_0 = 1 et u_{n+1} = 2u_n + 3. Quelle est la forme explicite de u_n ?","option_a":"A. u_n = 2^n - 3","option_b":"B. u_n = 4*2^n - 3","option_c":"C. u_n = 2*2^n - 3","option_d":"D. u_n = 3*2^n - 2","option_e":"","option_f":"","bonne_reponse":"c","explication":"En résolvant l'équation de récurrence, on trouve u_n = 4*2^n - 3 (solution générale de l'équation homogène + solution particulière).","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. u_n = 2^n - 3\", \"b\": \"B. u_n = 4*2^n - 3\", \"c\": \"C. u_n = 2*2^","_debug_options_count":4},{"id":48735,"question":"La suite 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, elle n'a pas de limite finie, donc elle est 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":48736,"question":"Quel théorème permet d'affirmer que si u_n ≤ v_n pour tout n et que u_n → L, alors v_n → L' avec L ≤ L' ?","option_a":"A. Théorème de comparaison","option_b":"B. Théorème des gendarmes","option_c":"C. Théorème de Bolzano-Weierstrass","option_d":"D. Théorème de la limite monotone","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème de comparaison stipule que si u_n ≤ v_n et u_n → L, alors v_n → L' avec L ≤ L'.","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. Théorème de comparaison\", \"b\": \"B. Théorème des gendarmes\"","_debug_options_count":4},{"id":48737,"question":"La suite u_n = n^2 - 5n + 1 est-elle minorée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette suite est une fonction du second degré qui tend vers +∞, donc elle est minorée par son minimum (ici, pour n=3, u_3 = -3).","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":48738,"question":"Quelle est la limite de la suite u_n = (n^2 + 1)\/(n + 1) - n ?","option_a":"A. 0","option_b":"B. +∞","option_c":"C. -1","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"d","explication":"En simplifiant l'expression, on obtient u_n = (n^2 + 1 - n^2 - n)\/(n + 1) = (1 - n)\/(n + 1) → -1.","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. 0\", \"b\": \"B. +∞\", \"c\": \"C. -1\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":48739,"question":"Une suite bornée est-elle toujours convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite bornée n'est pas nécessairement convergente (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":48740,"question":"Soit la suite u_n = 1\/n. Que vaut lim_{n→+∞} u_n ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite u_n = 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\": \"a\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":48741,"question":"Le théorème de Bolzano-Weierstrass s'applique-t-il aux suites bornées ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème de Bolzano-Weierstrass affirme que toute suite bornée admet une sous-suite 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}]