Quiz — Livre 02 L (المتتاليات).pdf

Question 1 sur 10 20:00

[{"id":67856,"question":"Quelle est la limite de la suite définie par uₙ = (2n + 1)\/(n + 3) quand n tend vers +∞ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = (2 + 1\/n)\/(1 + 3\/n). Quand n → +∞, 1\/n → 0, donc uₙ → 2\/1 = 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":67857,"question":"Une suite croissante et non majorée est toujours divergente vers +∞.","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 bien vers +∞ (théorème de la limite monotone).","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":67858,"question":"Quelle est la nature de la série de terme général uₙ = 1\/n² ?","option_a":"A. Convergente","option_b":"B. Divergente","option_c":"C. Indéterminée","option_d":"D. Oscillante","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série Σ1\/n² est une série de Riemann convergente (car 2 \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\": \"A. Convergente\", \"b\": \"B. Divergente\", \"c\": \"C. Indéterminée\", ","_debug_options_count":4},{"id":67859,"question":"Soit (uₙ) une suite définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 1. Quelle est la formule explicite de uₙ ?","option_a":"A. uₙ = 2ⁿ","option_b":"B. uₙ = 2ⁿ - 1","option_c":"C. uₙ = 2ⁿ + 1","option_d":"D. uₙ = n + 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"En résolvant la récurrence linéaire, on trouve uₙ = 2ⁿ - 1 (solution particulière + solution homogène).","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. uₙ = 2ⁿ\", \"b\": \"B. uₙ = 2ⁿ - 1\", \"c\": \"C. uₙ = 2ⁿ ","_debug_options_count":4},{"id":67860,"question":"La suite uₙ = (-1)ⁿ est bornée mais ne converge pas.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite uₙ = (-1)ⁿ est bornée (par -1 et 1) mais alterne entre ces deux valeurs, donc elle n'a pas de 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":67861,"question":"Quelle est la somme de la série géométrique de premier terme 3 et de raison 1\/2 ?","option_a":"A. 3","option_b":"B. 6","option_c":"C. +∞","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"b","explication":"La somme d'une série géométrique convergente est S = premier terme \/ (1 - raison) = 3 \/ (1 - 1\/2) = 6.","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. 3\", \"b\": \"B. 6\", \"c\": \"C. +∞\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":67862,"question":"Soit (uₙ) une suite telle que uₙ₊₁ - uₙ = 1\/n² pour tout n ≥ 1. Que peut-on dire de la convergence de (uₙ) ?","option_a":"A. Elle converge","option_b":"B. Elle diverge","option_c":"C. On ne peut pas conclure","option_d":"D. Elle oscille","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série Σ1\/n² converge (série de Riemann), donc la suite (uₙ) converge (théorème de comparaison des séries).","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. Elle converge\", \"b\": \"B. Elle diverge\", \"c\": \"C. On ne peut pa","_debug_options_count":4},{"id":67863,"question":"La suite uₙ = n² - 5n + 6 est-elle croissante à partir d'un certain rang ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction associée f(x) = x² - 5x + 6 est f'(x) = 2x - 5. Pour x \u003E 2.5, f'(x) \u003E 0, donc la suite est croissante à partir de n = 3.","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":67864,"question":"Quelle est la limite de la suite uₙ = (1 + 1\/n)ⁿ quand n tend vers +∞ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. e","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"Cette suite est la définition même du nombre e (≈ 2.718).","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. e\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":67865,"question":"Soit (uₙ) une suite telle que uₙ₊₂ = uₙ₊₁ + uₙ pour tout n ≥ 0 (suite de Fibonacci). Si u₀ = 0 et u₁ = 1, quelle est la valeur de u₅ ?","option_a":"A. 3","option_b":"B. 5","option_c":"C. 8","option_d":"D. 13","option_e":"","option_f":"","bonne_reponse":"b","explication":"Calculons les termes : u₂ = 1, u₃ = 2, u₄ = 3, u₅ = 5. La suite de Fibonacci est définie par uₙ₊₂ = uₙ₊₁ + uₙ.","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. 3\", \"b\": \"B. 5\", \"c\": \"C. 8\", \"d\": \"D. 13\"}}","_debug_options_count":4}]