Quiz — Cours suites réelles.pdf

Question 1 sur 10 20:00

[{"id":75766,"question":"Quelle est la limite de la suite définie par u_n = (2n + 1)\/(n + 3) lorsque n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n, on obtient (2 + 1\/n)\/(1 + 3\/n) → 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":75767,"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 un théorème fondamental : 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":75768,"question":"Soit la suite définie par u_0 = 1 et u_{n+1} = 2u_n + 1. Que vaut u_2 ?","option_a":"A. 3","option_b":"B. 5","option_c":"C. 7","option_d":"D. 9","option_e":"","option_f":"","bonne_reponse":"c","explication":"u_1 = 2*1 + 1 = 3, u_2 = 2*3 + 1 = 7.","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. 3\", \"b\": \"B. 5\", \"c\": \"C. 7\", \"d\": \"D. 9\"}}","_debug_options_count":4},{"id":75769,"question":"La suite v_n = (-1)^n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette suite oscille entre -1 et 1, elle n'a pas de limite finie : elle est 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":75770,"question":"Quelle méthode utiliser pour étudier la convergence d'une suite définie par récurrence ?","option_a":"A. Calculer les 100 premiers termes","option_b":"B. Étudier la fonction associée","option_c":"C. Appliquer le théorème des suites adjacentes","option_d":"D. Utiliser le théorème de la limite monotone","option_e":"","option_f":"","bonne_reponse":"d","explication":"Le théorème de la limite monotone s'applique si la suite est monotone et bornée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A. Calculer les 100 premiers termes\", \"b\": \"B. Étudier la foncti","_debug_options_count":4},{"id":75771,"question":"Soit u_n = n\/(n^2 + 1). Que peut-on dire de sa limite ?","option_a":"A. Elle tend vers 0","option_b":"B. Elle tend vers 1","option_c":"C. Elle tend vers +∞","option_d":"D. Elle n'a pas de limite","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant par n^2, on obtient 1\/(n + 1\/n) → 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. Elle tend vers 0\", \"b\": \"B. Elle tend vers 1\", \"c\": \"C. Elle t","_debug_options_count":4},{"id":75772,"question":"Une suite décroissante et minorée est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un théorème : une suite décroissante et minorée converge vers sa borne inférieure.","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":75773,"question":"Quelle est la nature de la suite définie par u_n = 3 - 1\/n ?","option_a":"A. Croissante et convergente vers 3","option_b":"B. Décroissante et convergente vers 3","option_c":"C. Constante","option_d":"D. Divergente","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est croissante (car 1\/n décroît) et converge vers 3.","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 et convergente vers 3\", \"b\": \"B. Décroissante et c","_debug_options_count":4},{"id":75774,"question":"Le théorème des gendarmes s'applique-t-il si une suite est encadrée par deux suites divergentes ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème des gendarmes nécessite que les deux suites encadrantes convergent vers la même limite.","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":75775,"question":"Soit u_n = (n^2 + 1)\/(2n^2 - 3). Que vaut sa limite ?","option_a":"A. 0","option_b":"B. 1\/2","option_c":"C. 1","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant par n^2, on obtient (1 + 1\/n^2)\/(2 - 3\/n^2) → 1\/2.","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\/2\", \"c\": \"C. 1\", \"d\": \"D. +∞\"}}","_debug_options_count":4}]