Quiz — Fiche méthodes - Suites réelles.pdf

Question 1 sur 10 20:00

[{"id":34920,"question":"Quelle est la limite de la suite définie par u_n = (-1)^n pour n ∈ ℕ ?","option_a":"0","option_b":"1","option_c":"La suite n'a pas de limite","option_d":"-1","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite alterne entre -1 et 1, elle n'a donc pas de limite (elle est divergente).","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\": \"La suite n'a pas de limite\", \"d\": \"-1\"}}","_debug_options_count":4},{"id":34921,"question":"Soit une suite (u_n) telle que u_n = n\/(n+1). Que peut-on dire de sa limite ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En simplifiant, u_n = 1\/(1 + 1\/n), donc lim(u_n) = 1 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\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":34922,"question":"Quelle méthode permet de montrer qu'une suite est croissante ?","option_a":"Étudier le signe de u_{n+1} - u_n","option_b":"Calculer la dérivée de u_n","option_c":"Comparer u_n à une suite connue","option_d":"Trouver un majorant","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour montrer qu'une suite est croissante, on étudie le signe de la différence u_{n+1} - u_n.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Étudier le signe de u_{n+1} - u_n\", \"b\": \"Calculer la dérivée ","_debug_options_count":4},{"id":34923,"question":"Soit la suite définie par u_0 = 2 et u_{n+1} = √(u_n + 3). Que peut-on dire de sa limite ?","option_a":"La suite diverge vers +∞","option_b":"La suite converge vers 3","option_c":"La suite converge vers -3","option_d":"La limite n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"En supposant que la suite converge vers L, on a L = √(L + 3), donc L = 3 (car L ≥ 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"La suite diverge vers +∞\", \"b\": \"La suite converge vers 3\", \"c\"","_debug_options_count":4},{"id":34924,"question":"Le théorème des gendarmes s'applique-t-il si (v_n) ≤ (u_n) ≤ (w_n) et lim(v_n) = lim(w_n) = L ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, si (v_n) et (w_n) convergent vers la même limite L, alors (u_n) 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\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":34925,"question":"Quelle est la limite de la suite u_n = n^2 \/ (n^2 + 1) ?","option_a":"0","option_b":"1","option_c":"1\/2","option_d":"La suite diverge","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n^2, on obtient u_n = 1 \/ (1 + 1\/n^2), donc 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\": \"1\/2\", \"d\": \"La suite diverge\"}}","_debug_options_count":4},{"id":34926,"question":"Soit une suite (u_n) définie par u_{n+1} = 2u_n - 1 avec u_0 = 3. Que vaut u_2 ?","option_a":"3","option_b":"5","option_c":"7","option_d":"9","option_e":"","option_f":"","bonne_reponse":"c","explication":"u_1 = 2*3 - 1 = 5, u_2 = 2*5 - 1 = 9.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"3\", \"b\": \"5\", \"c\": \"7\", \"d\": \"9\"}}","_debug_options_count":4},{"id":34927,"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":"Non, une suite bornée peut être divergente (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":34928,"question":"Quelle est la limite de la suite u_n = (3n + 1)\/(2n - 5) ?","option_a":"3\/2","option_b":"1","option_c":"0","option_d":"La suite diverge","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n, on obtient u_n = (3 + 1\/n)\/(2 - 5\/n), donc lim(u_n) = 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\": \"3\/2\", \"b\": \"1\", \"c\": \"0\", \"d\": \"La suite diverge\"}}","_debug_options_count":4},{"id":34929,"question":"Soit (u_n) une suite telle que u_n = n\/(n^2 + 1). Que peut-on dire de sa limite ?","option_a":"0","option_b":"1","option_c":"1\/2","option_d":"La suite diverge","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n^2, on obtient u_n = 1\/(n + 1\/n), donc lim(u_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\": \"0\", \"b\": \"1\", \"c\": \"1\/2\", \"d\": \"La suite diverge\"}}","_debug_options_count":4}]