Quiz — 13_ComparaisonSuites.pdf

Question 1 sur 10 20:00

[{"id":87253,"question":"Soient deux suites (uₙ) et (vₙ) telles que pour tout n ≥ 0, uₙ ≤ vₙ. Si lim(uₙ) = +∞, que peut-on dire de lim(vₙ) ?","option_a":"A. lim(vₙ) = +∞","option_b":"B. lim(vₙ) = -∞","option_c":"C. On ne peut rien dire","option_d":"D. lim(vₙ) = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si uₙ ≤ vₙ et lim(uₙ) = +∞, alors par comparaison, lim(vₙ) = +∞ également.","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. lim(vₙ) = +∞\", \"b\": \"B. lim(vₙ) = -∞\", \"c\": \"C. On ne ","_debug_options_count":4},{"id":87254,"question":"Le théorème des gendarmes s'applique uniquement si les deux suites encadrantes convergent vers la même limite.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","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\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":87255,"question":"Soit (uₙ) une suite croissante et non majorée. Que peut-on affirmer sur sa limite ?","option_a":"A. lim(uₙ) = +∞","option_b":"B. lim(uₙ) = 0","option_c":"C. lim(uₙ) n'existe pas","option_d":"D. lim(uₙ) = -∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite croissante et non majorée diverge vers +∞.","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. lim(uₙ) = +∞\", \"b\": \"B. lim(uₙ) = 0\", \"c\": \"C. lim(uₙ)","_debug_options_count":4},{"id":87256,"question":"Si pour tout n, 0 ≤ uₙ ≤ 1\/n, alors lim(uₙ) = ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. On ne peut pas conclure","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par le théorème des gendarmes, comme 0 ≤ uₙ ≤ 1\/n et lim(1\/n) = 0, alors lim(uₙ) = 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\", \"c\": \"C. +∞\", \"d\": \"D. On ne peut pas concl","_debug_options_count":4},{"id":87257,"question":"Une suite minorée est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite minorée n'est pas nécessairement convergente (exemple : uₙ = 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":87258,"question":"Soient (uₙ) et (vₙ) deux suites telles que uₙ = n² et vₙ = n³. Que peut-on dire de leur comparaison ?","option_a":"A. uₙ ≤ vₙ pour tout n","option_b":"B. vₙ ≤ uₙ pour tout n","option_c":"C. uₙ = vₙ pour tout n","option_d":"D. Aucune comparaison possible","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour tout n ≥ 1, n² ≤ n³, donc uₙ ≤ vₙ.","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. uₙ ≤ vₙ pour tout n\", \"b\": \"B. vₙ ≤ uₙ pour tout n","_debug_options_count":4},{"id":87259,"question":"Si (uₙ) est une suite décroissante et minorée, alors elle est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par le théorème de la limite monotone, une suite décroissante et minorée est convergente.","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":87260,"question":"Soit (uₙ) une suite telle que uₙ = (-1)ⁿ \/ n. Que vaut lim(uₙ) ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. -1","option_d":"D. La suite diverge","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par le théorème des gendarmes, comme |uₙ| = 1\/n → 0, alors lim(uₙ) = 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\", \"c\": \"C. -1\", \"d\": \"D. La suite diverge\"}}","_debug_options_count":4},{"id":87261,"question":"Si lim(uₙ) = L et lim(vₙ) = L, alors lim(uₙ + vₙ) = 2L.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par les propriétés des limites, lim(uₙ + vₙ) = lim(uₙ) + lim(vₙ) = L + L = 2L.","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":87262,"question":"Soit (uₙ) une suite telle que uₙ = n + (-1)ⁿ. Que peut-on dire de sa limite ?","option_a":"A. lim(uₙ) = +∞","option_b":"B. lim(uₙ) = -∞","option_c":"C. La suite diverge","option_d":"D. lim(uₙ) = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"uₙ = n + (-1)ⁿ ≥ n - 1 → lim(uₙ) = +∞ par comparaison.","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. lim(uₙ) = +∞\", \"b\": \"B. lim(uₙ) = -∞\", \"c\": \"C. La sui","_debug_options_count":4}]