Quiz — Suites reélle 2.pdf

Question 1 sur 5 10:00

[{"id":2235,"question":"Soit la suite définie par uₙ = (2n + 1)\/(n + 3). Quelle est sa limite quand n tend vers +∞ ?","option_a":"0","option_b":"1","option_c":"2","option_d":"3","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant le numérateur et le dénominateur par n, on obtient uₙ = (2 + 1\/n)\/(1 + 3\/n). Quand n → +∞, 1\/n → 0, donc uₙ → 2\/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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"3\"}}","_debug_options_count":4},{"id":2236,"question":"La suite vₙ = (-1)ⁿ est-elle convergente ?","option_a":"Oui, car elle est bornée","option_b":"Non, car elle n'a pas de limite","option_c":"Oui, car elle est périodique","option_d":"Non, car elle est croissante","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite vₙ alterne entre -1 et 1 et n'admet pas de limite unique quand n → +∞. Elle est donc divergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Oui, car elle est bornée\", \"b\": \"Non, car elle n'a pas de limite","_debug_options_count":4},{"id":2237,"question":"Si pour tout n, uₙ ≤ vₙ ≤ wₙ et que lim uₙ = lim wₙ = L, alors que peut-on dire de la suite vₙ ?","option_a":"Elle diverge","option_b":"Elle converge vers L","option_c":"Elle est bornée mais pas convergente","option_d":"Elle est croissante","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est le théorème des gendarmes : si vₙ est encadrée par deux suites convergentes vers L, alors vₙ converge aussi vers L.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Elle diverge\", \"b\": \"Elle converge vers L\", \"c\": \"Elle est borné","_debug_options_count":4},{"id":2238,"question":"Soit la suite uₙ = n² - 5n + 6. Quelle est sa limite quand n tend vers +∞ ?","option_a":"-∞","option_b":"+∞","option_c":"0","option_d":"5","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le terme dominant est n², donc uₙ → +∞ quand 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\": \"-∞\", \"b\": \"+∞\", \"c\": \"0\", \"d\": \"5\"}}","_debug_options_count":4},{"id":2239,"question":"La suite uₙ = 1 + 1\/2 + 1\/4 + ... + 1\/2ⁿ est-elle convergente ? Si oui, vers quelle valeur ?","option_a":"Non, elle diverge","option_b":"Oui, vers 1","option_c":"Oui, vers 2","option_d":"Oui, vers 0","option_e":"","option_f":"","bonne_reponse":"c","explication":"C'est une suite géométrique de raison 1\/2. Sa somme est Sₙ = 2(1 - 1\/2ⁿ⁺¹), donc lim Sₙ = 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\": \"Non, elle diverge\", \"b\": \"Oui, vers 1\", \"c\": \"Oui, vers 2\", \"d\": ","_debug_options_count":4}]