Quiz — Bibmath28 Séries entières ( agrega & 11 exes).pdf

Question 1 sur 10 20:00

[{"id":47368,"question":"Quel est le rayon de convergence de la série entière Σ (n! \/ nⁿ) xⁿ ?","option_a":"A. 0","option_b":"B. e","option_c":"C. +∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"En appliquant le critère de d'Alembert : lim |aₙ₊₁ \/ aₙ| = lim (nⁿ \/ (n+1)ⁿ⁺¹) * (n+1) = lim (n \/ (n+1))ⁿ = 1\/e. Le rayon de convergence est donc 1\/(1\/e) = e.","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. e\", \"c\": \"C. +∞\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":47369,"question":"La série entière Σ xⁿ \/ n² converge uniformément sur [-1, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La convergence est uniforme sur [-1, 1] car la série Σ 1\/n² converge (série de Riemann avec α=2\u003E1), et |xⁿ \/ n²| ≤ 1\/n² pour |x| ≤ 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":47370,"question":"Quelle est la somme de la série entière Σ (-1)ⁿ x²ⁿ⁺¹ \/ (2n+1) pour |x| \u003C 1 ?","option_a":"A. arctan(x)","option_b":"B. arctan(x²)","option_c":"C. x \/ (1 + x²)","option_d":"D. x arctan(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série est la série de Taylor de arctan(x) : arctan(x) = Σ (-1)ⁿ x²ⁿ⁺¹ \/ (2n+1) pour |x| \u003C 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. arctan(x)\", \"b\": \"B. arctan(x²)\", \"c\": \"C. x \/ (1 + x²)\", \"d","_debug_options_count":4},{"id":47371,"question":"Le rayon de convergence de Σ n xⁿ est :","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. e","option_e":"","option_f":"","bonne_reponse":"b","explication":"En appliquant le critère de Cauchy : lim sup |aₙ|^(1\/n) = lim (n)^(1\/n) = 1. Le rayon de convergence est donc 1\/1 = 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. +∞\", \"d\": \"D. e\"}}","_debug_options_count":4},{"id":47372,"question":"La série entière Σ xⁿ \/ n converge simplement sur [-1, 1[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série converge simplement sur [-1, 1[ mais pas en x=1 (série harmonique divergente). La convergence est uniforme sur [-1, r] pour tout r \u003C 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":47373,"question":"Quelle est la dérivée de la série entière Σ xⁿ \/ n ! ?","option_a":"A. Σ xⁿ \/ (n-1) !","option_b":"B. Σ xⁿ⁻¹ \/ (n-1) !","option_c":"C. Σ n xⁿ⁻¹ \/ n !","option_d":"D. eˣ","option_e":"","option_f":"","bonne_reponse":"b","explication":"La série Σ xⁿ \/ n ! est la série de Taylor de eˣ. Sa dérivée est Σ n xⁿ⁻¹ \/ n ! = Σ xⁿ⁻¹ \/ (n-1) ! = eˣ.","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. Σ xⁿ \/ (n-1) !\", \"b\": \"B. Σ xⁿ⁻¹ \/ (n-1) !\", \"c\": \"C.","_debug_options_count":4},{"id":47374,"question":"Le rayon de convergence de Σ (ln n) xⁿ est :","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. e","option_e":"","option_f":"","bonne_reponse":"b","explication":"En appliquant le critère de Cauchy : lim sup (ln n)^(1\/n) = 1. Le rayon de convergence est donc 1\/1 = 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. +∞\", \"d\": \"D. e\"}}","_debug_options_count":4},{"id":47375,"question":"La série entière Σ (-1)ⁿ xⁿ converge simplement sur ]-1, 1[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série converge simplement sur ]-1, 1[ mais aussi en x=1 et x=-1 (série alternée convergente). La convergence est uniforme sur [-1, 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":47376,"question":"Quelle est l'intégrale terme à terme de Σ (-1)ⁿ x²ⁿ \/ (2n) ! pour |x| \u003C +∞ ?","option_a":"A. cos(x)","option_b":"B. sin(x)","option_c":"C. cosh(x)","option_d":"D. sinh(x)","option_e":"","option_f":"","bonne_reponse":"c","explication":"La série est la série de Taylor de cosh(x) = (eˣ + e⁻ˣ)\/2. Son intégrale terme à terme donne sinh(x) + C.","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. cos(x)\", \"b\": \"B. sin(x)\", \"c\": \"C. cosh(x)\", \"d\": \"D. sinh(x)","_debug_options_count":4},{"id":47377,"question":"Le rayon de convergence de Σ n! xⁿ est :","option_a":"A. 0","option_b":"B. 1","option_c":"C. e","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant le critère de d'Alembert : lim |aₙ₊₁ \/ aₙ| = lim (n+1) = +∞. Le rayon de convergence est donc 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. e\", \"d\": \"D. +∞\"}}","_debug_options_count":4}]