Quiz — 632f3387b239c_-_Suites réelles.pdf

Question 1 sur 10 20:00

[{"id":63429,"question":"Quelle est la limite de la suite u_n = 3 + 1\/n lorsque n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"3","option_d":"L'infini","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite u_n = 3 + 1\/n converge vers 3 car 1\/n tend vers 0 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\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"3\", \"d\": \"L'infini\"}}","_debug_options_count":4},{"id":63430,"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 diverge vers +∞, donc elle est divergente.","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":63431,"question":"Quelle est la raison d'une suite géométrique dont le premier terme est 5 et le troisième terme est 20 ?","option_a":"2","option_b":"4","option_c":"-2","option_d":"1\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le troisième terme est u_3 = u_1 * r^2 = 5 * r^2 = 20 ⇒ r^2 = 4 ⇒ r = 2 (car r \u003E 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\": \"2\", \"b\": \"4\", \"c\": \"-2\", \"d\": \"1\/2\"}}","_debug_options_count":4},{"id":63432,"question":"La suite u_n = (-1)^n est-elle bornée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite u_n = (-1)^n prend uniquement les valeurs -1 et 1, donc elle est bornée par -1 et 1.","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":63433,"question":"Quelle est la limite de la suite u_n = n^2 \/ (n+1) lorsque n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"L'infini","option_d":"n","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n^2, on obtient u_n = 1 \/ (1\/n + 1\/n^2) → 1 lorsque n → ∞.","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\": \"L'infini\", \"d\": \"n\"}}","_debug_options_count":4},{"id":63434,"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":"D'après le théorème de la limite monotone, 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":63435,"question":"Quelle est la formule du terme général d'une suite arithmétique de raison r et de premier terme u_0 ?","option_a":"u_n = u_0 + n","option_b":"u_n = u_0 * r^n","option_c":"u_n = u_0 + n*r","option_d":"u_n = u_0 \/ n","option_e":"","option_f":"","bonne_reponse":"c","explication":"La formule du terme général d'une suite arithmétique est u_n = u_0 + n*r.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"u_n = u_0 + n\", \"b\": \"u_n = u_0 * r^n\", \"c\": \"u_n = u_0 + n*r\", \"","_debug_options_count":4},{"id":63436,"question":"La suite u_n = n \/ (n+1) est-elle convergente ? Si oui, quelle est sa limite ?","option_a":"Non, elle diverge","option_b":"Oui, limite = 0","option_c":"Oui, limite = 1","option_d":"Oui, limite = -1","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite u_n = n \/ (n+1) converge vers 1 car n\/(n+1) = 1 \/ (1 + 1\/n) → 1 lorsque n → ∞.","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, limite = 0\", \"c\": \"Oui, limite = 1","_debug_options_count":4},{"id":63437,"question":"Si une suite est bornée, alors elle est 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 (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":63438,"question":"Quelle est la limite de la suite u_n = (2n + 1) \/ (3n - 2) lorsque n tend vers l'infini ?","option_a":"0","option_b":"2\/3","option_c":"1","option_d":"L'infini","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n, on obtient u_n = (2 + 1\/n) \/ (3 - 2\/n) → 2\/3 lorsque 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\": \"0\", \"b\": \"2\/3\", \"c\": \"1\", \"d\": \"L'infini\"}}","_debug_options_count":4}]