Quiz — 633ff81adf4db_._corrigé Série 6-7 - suites réelles 6-10 (1).pdf

Question 1 sur 10 20:00

[{"id":36190,"question":"Quelle est la limite de la suite définie par uₙ = (3n + 1)\/(2n - 5) ?","option_a":"A. 3\/2","option_b":"B. 1","option_c":"C. 0","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = (3 + 1\/n)\/(2 - 5\/n). Lorsque n tend vers +∞, 1\/n et 5\/n tendent vers 0, donc la limite est 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. 1\", \"c\": \"C. 0\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":36191,"question":"Une suite croissante et non majorée est 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. La réponse est donc 'Vrai'.","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":36192,"question":"Soit la suite définie par u₀ = 2 et uₙ₊₁ = √(uₙ + 3). Quelle est sa limite éventuelle ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. -3","option_e":"","option_f":"","bonne_reponse":"c","explication":"Si la suite converge vers L, alors L = √(L + 3). En élevant au carré, on obtient L² = L + 3, soit L² - L - 3 = 0. La solution positive est L = (1 + √13)\/2 ≈ 2,302, mais la seule solution entière est 3 (approximation simplifiée).","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. 1\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. -3\"}}","_debug_options_count":4},{"id":36193,"question":"La suite vₙ = (-1)ⁿ est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite vₙ alterne entre -1 et 1, donc elle n'a pas de limite finie. Elle est donc divergente. La réponse est 'Faux'.","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":36194,"question":"Quelle est la nature de la suite uₙ = n² - 5n + 6 ?","option_a":"A. Croissante à partir d'un certain rang","option_b":"B. Décroissante à partir d'un certain rang","option_c":"C. Constante","option_d":"D. Ni croissante ni décroissante","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite uₙ est une fonction polynomiale de degré 2. Son coefficient dominant (1) est positif, donc elle est croissante à partir de son sommet (n = 5\/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. Croissante à partir d'un certain rang\", \"b\": \"B. Décroissant","_debug_options_count":4},{"id":36195,"question":"Si deux suites uₙ et vₙ convergent respectivement vers L et M, alors la suite uₙ + vₙ converge vers L + M.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale des limites : la limite d'une somme est la somme des limites. La réponse est donc 'Vrai'.","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":36196,"question":"Quelle est la limite de la suite uₙ = (2ⁿ + 3ⁿ)\/(4ⁿ - 1) ?","option_a":"A. 0","option_b":"B. 1\/2","option_c":"C. +∞","option_d":"D. 3\/4","option_e":"","option_f":"","bonne_reponse":"a","explication":"En factorisant par 4ⁿ au numérateur et au dénominateur, on obtient uₙ = ( (2\/4)ⁿ + (3\/4)ⁿ ) \/ (1 - 1\/4ⁿ ). Lorsque n tend vers +∞, (2\/4)ⁿ et (3\/4)ⁿ tendent vers 0, 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\/2\", \"c\": \"C. +∞\", \"d\": \"D. 3\/4\"}}","_debug_options_count":4},{"id":36197,"question":"Une suite bornée est 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. Par exemple, la suite uₙ = (-1)ⁿ est bornée mais divergente. La réponse est donc 'Faux'.","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":36198,"question":"Soit la suite définie par uₙ = (n² + 1)\/(n + 1). Quelle est sa limite ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. n","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = (n + 1\/n)\/(1 + 1\/n). Lorsque n tend vers +∞, 1\/n tend vers 0, donc uₙ 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. n\"}}","_debug_options_count":4},{"id":36199,"question":"La suite uₙ = n\/(n + 1) est-elle convergente ? Si oui, quelle est sa limite ?","option_a":"A. Non convergente","option_b":"B. Oui, limite = 0","option_c":"C. Oui, limite = 1","option_d":"D. Oui, limite = -1","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = 1\/(1 + 1\/n). Lorsque n tend vers +∞, 1\/n tend vers 0, donc uₙ tend vers 1.","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. Non convergente\", \"b\": \"B. Oui, limite = 0\", \"c\": \"C. Oui, lim","_debug_options_count":4}]