Quiz — Magazine 12 Suites réelles.pdf

Question 1 sur 10 20:00

[{"id":12448,"question":"Quelle est la définition d’une suite convergente ?","option_a":"Une suite dont tous les termes sont égaux","option_b":"Une suite qui admet une limite finie","option_c":"Une suite qui tend vers l’infini","option_d":"Une suite dont les termes sont tous positifs","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite convergente est une suite qui admet une limite finie 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\": \"Une suite dont tous les termes sont égaux\", \"b\": \"Une suite qui ","_debug_options_count":4},{"id":12449,"question":"Le théorème de Bolzano-Weierstrass s’applique-t-il à toutes les suites ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème de Bolzano-Weierstrass s’applique uniquement aux suites bornées.","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":12450,"question":"Si une suite est croissante et majorée, alors elle est convergente. Vrai ou Faux ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite croissante et majorée est convergente (théorème de convergence 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":12451,"question":"Quelle est la limite de la suite définie par u_n = (2n + 1)\/(n + 3) ?","option_a":"0","option_b":"1","option_c":"2","option_d":"3","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n, on obtient lim(u_n) = 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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"3\"}}","_debug_options_count":4},{"id":12452,"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":12453,"question":"Quelle propriété vérifie une suite géométrique de raison q avec |q| \u003C 1 ?","option_a":"Elle est croissante","option_b":"Elle est décroissante","option_c":"Elle est convergente","option_d":"Elle est divergente","option_e":"","option_f":"","bonne_reponse":"c","explication":"Une suite géométrique de raison q avec |q| \u003C 1 est convergente vers 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Elle est croissante\", \"b\": \"Elle est décroissante\", \"c\": \"Elle e","_debug_options_count":4},{"id":12454,"question":"Si u_n ≤ v_n pour tout n et lim(u_n) = lim(v_n) = L, alors lim(w_n) = L où w_n est une suite intermédiaire. Vrai ou Faux ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C’est le théorème des gendarmes : si u_n ≤ w_n ≤ v_n, alors lim(w_n) = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":12455,"question":"Quelle est la limite de la suite u_n = n^2 \/ (n^2 + 1) ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"1\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n^2, on obtient lim(u_n) = 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\": \"0\", \"b\": \"1\", \"c\": \"∞\", \"d\": \"1\/2\"}}","_debug_options_count":4},{"id":12456,"question":"Une suite peut-elle être à la fois croissante et décroissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite constante est à la fois croissante et décroissante.","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":12457,"question":"Quelle est la condition nécessaire pour qu’une suite soit convergente ?","option_a":"Elle doit être bornée","option_b":"Elle doit être monotone","option_c":"Elle doit être définie pour tout n","option_d":"Elle doit être croissante","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite convergente doit être bornée (condition nécessaire mais non suffisante).","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 doit être bornée\", \"b\": \"Elle doit être monotone\", \"c\": \"","_debug_options_count":4}]