Quiz — ds7-suites-limites.pdf

Question 1 sur 10 20:00

[{"id":22259,"question":"Quelle est la limite de la suite définie par uₙ = (3n² + 2n - 1)\/(2n² - n + 5) ?","option_a":"A. 0","option_b":"B. 3\/2","option_c":"C. +∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"On divise numérateur et dénominateur par n² (le terme dominant). On obtient uₙ = (3 + 2\/n - 1\/n²)\/(2 - 1\/n + 5\/n²). Quand n → +∞, les termes en 1\/n tendent vers 0, donc la limite est 3\/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. 3\/2\", \"c\": \"C. +∞\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":22260,"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 le théorème de convergence monotone : 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":22261,"question":"Quelle est la limite de la suite uₙ = (-1)ⁿ × n ?","option_a":"A. 0","option_b":"B. +∞","option_c":"C. -∞","option_d":"D. La suite n'a pas de limite","option_e":"","option_f":"","bonne_reponse":"d","explication":"La suite alterne entre des valeurs positives et négatives de plus en plus grandes en valeur absolue. Elle n'a donc pas de limite.","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. 0\", \"b\": \"B. +∞\", \"c\": \"C. -∞\", \"d\": \"D. La suite n'a pas ","_debug_options_count":4},{"id":22262,"question":"Si pour tout n, vₙ ≤ uₙ ≤ wₙ et si lim vₙ = lim wₙ = L, alors lim uₙ = L.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème des gendarmes : si les suites encadrantes convergent vers la même limite L, alors la suite encadrée 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":22263,"question":"Quelle est la limite de la suite définie par u₀ = 1 et uₙ₊₁ = √(uₙ + 2) ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"On montre que la suite est croissante et majorée par 2. Elle converge donc vers la solution de L = √(L + 2), soit L = 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":22264,"question":"La suite uₙ = n² - 5n + 6 est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette suite est un polynôme du second degré. Son terme dominant est n², donc lim uₙ = +∞. 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":22265,"question":"Quelle méthode utiliser pour calculer lim (√(n² + n) - n) ?","option_a":"A. Factoriser n²","option_b":"B. Multiplier par le conjugué","option_c":"C. Appliquer le théorème des gendarmes","option_d":"D. Utiliser la formule de Taylor","option_e":"","option_f":"","bonne_reponse":"b","explication":"On multiplie par le conjugué √(n² + n) + n pour lever l'indétermination ∞ - ∞. On obtient (n² + n - n²)\/(√(n² + n) + n) = n\/(√(n² + n) + 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\": \"A. Factoriser n²\", \"b\": \"B. Multiplier par le conjugué\", \"c\": \"","_debug_options_count":4},{"id":22266,"question":"Toute suite bornée est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Exemple : la suite uₙ = (-1)ⁿ est bornée mais n'a pas de limite. Une suite bornée n'est pas nécessairement convergente.","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":22267,"question":"Quelle est la limite de la suite uₙ = (2ⁿ + 3ⁿ)\/(5ⁿ - 1) ?","option_a":"A. 0","option_b":"B. 3\/5","option_c":"C. +∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"On factorise par 3ⁿ au numérateur et 5ⁿ au dénominateur : uₙ = ( (2\/3)ⁿ + 1 ) \/ ( (5\/3)ⁿ - (1\/3)ⁿ ). Quand n → +∞, (2\/3)ⁿ et (1\/3)ⁿ tendent vers 0, donc la limite est 1\/(+∞) = 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. 3\/5\", \"c\": \"C. +∞\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":22268,"question":"Si une suite est 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":"C'est le théorème de convergence 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}]