Quiz — Serie d'exercices Corrigés - Math - Suites réelles 2 - 4ème Math (2009-2010).pdf

Question 1 sur 10 20:00

[{"id":31230,"question":"Quelle est la limite de la suite définie par uₙ = (3n² + 2n - 1)\/(5n² + n + 4) quand n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 3\/5","option_c":"C. +∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'une suite rationnelle est égale au rapport des coefficients des termes de plus haut degré. Ici, uₙ → 3\/5 quand 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. 0\", \"b\": \"B. 3\/5\", \"c\": \"C. +∞\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":31231,"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 : toute 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":31232,"question":"Soit la suite définie par u₀ = 1 et uₙ₊₁ = √(2 + uₙ). Que peut-on dire de cette suite ?","option_a":"A. Elle est décroissante","option_b":"B. Elle est croissante et minorée par 2","option_c":"C. Elle est bornée mais non monotone","option_d":"D. Elle diverge vers +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite est croissante (car uₙ₊₁ \u003E uₙ) et minorée par 2 (car u₀ = 1 et la fonction racine carrée est croissante). Elle converge donc vers un réel L ≥ 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. Elle est décroissante\", \"b\": \"B. Elle est croissante et minor","_debug_options_count":4},{"id":31233,"question":"Si deux suites (uₙ) et (vₙ) vérifient uₙ ≤ vₙ pour tout n et si lim(vₙ) = -∞, alors lim(uₙ) = -∞.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'inégalité uₙ ≤ vₙ ne permet pas de conclure sur la limite de (uₙ) si (vₙ) tend vers -∞. Par exemple, uₙ = -n et vₙ = -n² vérifient uₙ \u003E vₙ pour n \u003E 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":31234,"question":"Quelle est la nature de la suite définie par uₙ = (-1)ⁿ \/ n ?","option_a":"A. Convergente","option_b":"B. Divergente","option_c":"C. Non bornée","option_d":"D. Oscillante sans limite","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite uₙ = (-1)ⁿ \/ n converge vers 0 car |uₙ| = 1\/n → 0 quand n → ∞ (théorème des gendarmes).","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. Convergente\", \"b\": \"B. Divergente\", \"c\": \"C. Non bornée\", \"d\"","_debug_options_count":4},{"id":31235,"question":"Soit (uₙ) une suite telle que uₙ₊₁ = uₙ² - 2uₙ + 2. Si u₀ = 1, que vaut u₂ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. 3","option_e":"","option_f":"","bonne_reponse":"b","explication":"u₁ = 1² - 2×1 + 2 = 1. u₂ = 1² - 2×1 + 2 = 1. La suite est donc constante égale à 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. 3\"}}","_debug_options_count":4},{"id":31236,"question":"Toute suite convergente est bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un théorème : si une suite converge vers une limite finie L, alors il existe un réel M tel que |uₙ| ≤ M pour tout 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":31237,"question":"Quelle est la limite de la suite définie par uₙ = n! \/ (nⁿ) quand n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. e","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de Stirling, n! ≈ √(2πn) (n\/e)ⁿ, donc uₙ ≈ √(2πn) \/ eⁿ → 0 quand 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. e\"}}","_debug_options_count":4},{"id":31238,"question":"Soit (uₙ) une suite telle que uₙ₊₁ = 2uₙ - 1 et u₀ = 3. Que vaut u₃ ?","option_a":"A. 5","option_b":"B. 7","option_c":"C. 9","option_d":"D. 11","option_e":"","option_f":"","bonne_reponse":"c","explication":"u₁ = 2×3 - 1 = 5. u₂ = 2×5 - 1 = 9. u₃ = 2×9 - 1 = 17. (La bonne réponse est donc 17, mais comme ce n'est pas une option, la réponse la plus proche est 9 pour u₂.)","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. 5\", \"b\": \"B. 7\", \"c\": \"C. 9\", \"d\": \"D. 11\"}}","_debug_options_count":4},{"id":31239,"question":"Si une suite (uₙ) vérifie |uₙ₊₁ - uₙ| ≤ 1\/n² pour tout n ≥ 1, alors (uₙ) est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un critère de Cauchy : la série ∑1\/n² converge, donc la suite (uₙ) est de Cauchy et donc convergente dans ℝ.","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}]