Quiz — éxercices_corriges_suites_réelles.pdf

Question 1 sur 10 20:00

[{"id":63853,"question":"Quelle est la limite de la suite définie par u_n = (2n + 1)\/(3n - 5) quand n tend vers l'infini ?","option_a":"A) 0","option_b":"B) 2\/3","option_c":"C) 1","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)\/(3 - 5\/n). Quand n→∞, 1\/n et 5\/n tendent vers 0, donc la limite est 2\/3.","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\/3\", \"c\": \"C) 1\", \"d\": \"D) +∞\"}}","_debug_options_count":4},{"id":63854,"question":"Une suite croissante et non majorée est toujours divergente vers +∞.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : une suite croissante et non majorée n'a pas de borne supérieure, donc elle tend nécessairement vers +∞.","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":63855,"question":"Soit (u_n) une suite telle que pour tout n, u_{n+1} = 0.8 * u_n + 3. Si u_0 = 5, quelle est la limite de (u_n) ?","option_a":"A) 0","option_b":"B) 3","option_c":"C) 5","option_d":"D) 15","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite converge vers la solution de L = 0.8L + 3, soit L = 15. C'est une suite arithmético-géométrique 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\": \"A) 0\", \"b\": \"B) 3\", \"c\": \"C) 5\", \"d\": \"D) 15\"}}","_debug_options_count":4},{"id":63856,"question":"La suite définie par u_n = (-1)^n est-elle bornée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : pour tout n, -1 ≤ u_n ≤ 1, donc la suite est bornée.","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":63857,"question":"Quelle propriété caractérise une suite géométrique de raison q = 1 ?","option_a":"A) Constante","option_b":"B) Arithmétique","option_c":"C) Croissante","option_d":"D) Décroissante","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si q = 1, alors u_{n+1} = u_n pour tout n : 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\": \"a\", \"options\": {\"a\": \"A) Constante\", \"b\": \"B) Arithmétique\", \"c\": \"C) Croissante\", \"d\"","_debug_options_count":4},{"id":63858,"question":"Soit (u_n) une suite telle que u_n = n^2 + 3n + 2. Cette suite est-elle minorée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : pour tout n, u_n ≥ 2 (car n^2 + 3n ≥ 0 pour n ≥ 0). La suite est donc minorée par 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":63859,"question":"Quelle est la limite de la suite u_n = (n^2 + 1)\/(2n^2 - 3) quand n tend vers l'infini ?","option_a":"A) 0","option_b":"B) 1\/2","option_c":"C) 1","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n^2, on obtient u_n = (1 + 1\/n^2)\/(2 - 3\/n^2). Quand n→∞, la limite est 1\/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\": \"A) 0\", \"b\": \"B) 1\/2\", \"c\": \"C) 1\", \"d\": \"D) +∞\"}}","_debug_options_count":4},{"id":63860,"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 = (-1)^n \/ n converge vers 0 mais n'est pas monotone.","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":63861,"question":"Soit (u_n) une suite définie par u_0 = 1 et u_{n+1} = √(u_n + 2). Cette suite est-elle croissante ?","option_a":"A) Oui, car √(x+2) est croissante","option_b":"B) Non, car u_1 = √3 \u003C u_0 = 1","option_c":"C) Oui, car u_1 \u003E u_0","option_d":"D) On ne peut pas savoir","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = √(x+2) est croissante, donc si u_n \u003C u_{n+1}, alors f(u_n) \u003C f(u_{n+1}), soit u_{n+1} \u003C u_{n+2}. La suite est croissante.","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) Oui, car √(x+2) est croissante\", \"b\": \"B) Non, car u_1 = √","_debug_options_count":4},{"id":63862,"question":"Quelle est la limite de la suite u_n = (3^n + 2^n)\/(4^n) quand n tend vers l'infini ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) 3\/4","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En factorisant par 4^n au numérateur et dénominateur, on obtient u_n = ( (3\/4)^n + (2\/4)^n ). Les deux termes tendent vers 0 quand n→∞, 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) 1\", \"c\": \"C) 3\/4\", \"d\": \"D) +∞\"}}","_debug_options_count":4}]