Quiz — Suites monotones.pdf

Question 1 sur 10 20:00

[{"id":10413,"question":"Soit la suite définie par u_n = 3n - 2. Cette suite est :","option_a":"Croissante","option_b":"Décroissante","option_c":"Constante","option_d":"Non monotone","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite u_n = 3n - 2 est croissante car u_{n+1} - u_n = 3 \u003E 0 pour tout n.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Croissante\", \"b\": \"Décroissante\", \"c\": \"Constante\", \"d\": \"Non mo","_debug_options_count":4},{"id":10414,"question":"Une suite monotone non bornée est toujours divergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : une suite monotone non bornée est toujours divergente (vers +∞ ou -∞).","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":10415,"question":"Soit la suite définie par u_n = (-1)^n \/ n. Cette suite est :","option_a":"Croissante","option_b":"Décroissante","option_c":"Non monotone","option_d":"Constante","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite u_n = (-1)^n \/ n alterne entre valeurs positives et négatives, donc elle 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\": \"c\", \"options\": {\"a\": \"Croissante\", \"b\": \"Décroissante\", \"c\": \"Non monotone\", \"d\": \"Con","_debug_options_count":4},{"id":10416,"question":"Soit une suite croissante et majorée. Que peut-on dire de sa limite ?","option_a":"Elle est finie et égale à sa borne supérieure","option_b":"Elle est infinie","option_c":"Elle n'existe pas","option_d":"Elle est égale à 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite croissante et majorée converge vers sa borne supérieure (théorème de la limite monotone).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Elle est finie et égale à sa borne supérieure\", \"b\": \"Elle est","_debug_options_count":4},{"id":10417,"question":"La suite définie par u_n = n^2 est décroissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux : la suite u_n = n^2 est croissante car u_{n+1} - u_n = 2n + 1 \u003E 0 pour tout 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":10418,"question":"Soit la suite définie par u_n = 1\/n. Cette suite est :","option_a":"Croissante","option_b":"Décroissante","option_c":"Constante","option_d":"Non monotone","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite u_n = 1\/n est décroissante car u_{n+1} = 1\/(n+1) \u003C 1\/n = u_n pour tout 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\": \"Croissante\", \"b\": \"Décroissante\", \"c\": \"Constante\", \"d\": \"Non mo","_debug_options_count":4},{"id":10419,"question":"Une suite monotone peut-elle être bornée sans être convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux : une suite monotone et bornée est toujours convergente (théorème de la limite 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":10420,"question":"Soit la suite définie par u_n = 2n + 1. Sa limite est :","option_a":"0","option_b":"1","option_c":"+∞","option_d":"-∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite u_n = 2n + 1 tend vers +∞ lorsque n tend 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\": \"0\", \"b\": \"1\", \"c\": \"+∞\", \"d\": \"-∞\"}}","_debug_options_count":4},{"id":10421,"question":"La suite définie par u_n = (-2)^n est monotone.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux : la suite u_n = (-2)^n alterne entre valeurs positives et négatives, donc elle 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":10422,"question":"Soit une suite décroissante et minorée. Que peut-on dire de sa limite ?","option_a":"Elle est finie et égale à sa borne inférieure","option_b":"Elle est infinie","option_c":"Elle n'existe pas","option_d":"Elle est égale à 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite décroissante et minorée converge vers sa borne inférieure (théorème de la limite monotone).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Elle est finie et égale à sa borne inférieure\", \"b\": \"Elle est","_debug_options_count":4}]