Quiz — 4i-serie-suites_2018.pdf

Question 1 sur 10 20:00

[{"id":29230,"question":"Quelle est la limite de la suite définie par uₙ = n² - 5n + 3 lorsque n tend vers +∞ ?","option_a":"A. -∞","option_b":"B. +∞","option_c":"C. 0","option_d":"D. 3","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le terme dominant est n², qui tend vers +∞ lorsque n tend vers +∞. Les autres termes (-5n et +3) deviennent négligeables.","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. -∞\", \"b\": \"B. +∞\", \"c\": \"C. 0\", \"d\": \"D. 3\"}}","_debug_options_count":4},{"id":29231,"question":"Une suite croissante et majorée est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai d'après le théorème de la limite monotone : une suite croissante et majorée converge vers sa borne supérieure.","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":29232,"question":"Soit une suite géométrique de raison q = 0,5. Quelle est sa limite lorsque n tend vers +∞ ?","option_a":"A. 0","option_b":"B. +∞","option_c":"C. 1","option_d":"D. q","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une suite géométrique de raison |q| \u003C 1, la limite est 0 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\": \"a\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. +∞\", \"c\": \"C. 1\", \"d\": \"D. q\"}}","_debug_options_count":4},{"id":29233,"question":"La suite définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 1 est une suite arithmétique.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette suite est définie par récurrence et n'est pas de la forme uₙ₊₁ = uₙ + r. Elle est donc géométrique ou arithmético-géométrique, mais pas arithmétique.","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":29234,"question":"Quelle est la limite de la suite uₙ = (-1)ⁿ \/ n lorsque n tend vers +∞ ?","option_a":"A. -1","option_b":"B. 0","option_c":"C. +1","option_d":"D. n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le terme (-1)ⁿ oscille entre -1 et 1, mais est divisé par n qui tend vers +∞. La limite est donc 0 d'après le théorème des gendarmes.","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. -1\", \"b\": \"B. 0\", \"c\": \"C. +1\", \"d\": \"D. n'existe pas\"}}","_debug_options_count":4},{"id":29235,"question":"Le théorème des gendarmes s'applique uniquement aux suites croissantes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Le théorème des gendarmes s'applique à toute suite encadrée par deux suites convergentes vers la même limite, indépendamment de leur monotonie.","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":29236,"question":"Soit une suite définie par uₙ = 3 + 1\/n. Quelle est sa limite lorsque n tend vers +∞ ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le terme 1\/n tend vers 0 lorsque n tend vers +∞, donc uₙ tend vers 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. 3\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":29237,"question":"Une suite divergente peut avoir une limite infinie.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Une suite divergente peut tendre vers +∞ ou -∞, ce qui signifie qu'elle n'a pas de limite finie.","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":29238,"question":"Quelle est la limite de la suite uₙ = n \/ (n + 1) lorsque n tend vers +∞ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = 1 \/ (1 + 1\/n), qui tend vers 1 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\": \"b\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. n'existe pas\"}}","_debug_options_count":4},{"id":29239,"question":"Le théorème de la limite monotone s'applique aux suites strictement monotones.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Le théorème de la limite monotone s'applique aux suites monotones (croissantes ou décroissantes), qu'elles soient strictes ou non.","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}]