Quiz — Chap04_series_entieres.pdf

Question 1 sur 5 10:00

[{"id":1312,"question":"Quel est le critère de Cauchy pour déterminer le rayon de convergence R d'une série entière Σ aₙ xⁿ ?","option_a":"R = lim |aₙ₊₁ \/ aₙ|","option_b":"R = 1 \/ (lim sup |aₙ|^(1\/n))","option_c":"R = lim |aₙ|","option_d":"R = lim |aₙ₊₁ - aₙ|","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le critère de Cauchy stipule que le rayon de convergence R est l'inverse de la limite supérieure de |aₙ|^(1\/n). Cela permet de déterminer l'intervalle de convergence de la série.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"R = lim |aₙ₊₁ \/ aₙ|\", \"b\": \"R = 1 \/ (lim sup |aₙ|^(1\/n)","_debug_options_count":4},{"id":1313,"question":"Pour la série entière Σ (n! \/ (2n)!) xⁿ, quel est le rayon de convergence ?","option_a":"R = 0","option_b":"R = 2","option_c":"R = 1\/2","option_d":"R = ∞","option_e":"","option_f":"","bonne_reponse":"d","explication":"En appliquant le critère de d'Alembert, on trouve lim |aₙ₊₁ \/ aₙ| = lim (n+1)\/(2n+2) = 1\/2, mais comme la série converge pour tout x (car n! \/ (2n)! tend vers 0 très rapidement), le rayon de convergence est infini.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"R = 0\", \"b\": \"R = 2\", \"c\": \"R = 1\/2\", \"d\": \"R = ∞\"}}","_debug_options_count":4},{"id":1314,"question":"Quelle est la somme de la série entière Σ xⁿ pour |x| \u003C 1 ?","option_a":"1 \/ (1 - x)","option_b":"1 \/ (1 + x)","option_c":"x \/ (1 - x)","option_d":"x² \/ (1 - x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série géométrique Σ xⁿ converge vers 1 \/ (1 - x) pour |x| \u003C 1, une formule fondamentale en analyse et en probabilités.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"1 \/ (1 - x)\", \"b\": \"1 \/ (1 + x)\", \"c\": \"x \/ (1 - x)\", \"d\": \"x² \/","_debug_options_count":4},{"id":1315,"question":"Quel développement limité en 0 de la fonction f(x) = 1 \/ (1 - x) est valable pour |x| \u003C 1 ?","option_a":"1 + x + x² + x³ + ...","option_b":"1 - x + x² - x³ + ...","option_c":"x + x² \/ 2 + x³ \/ 3 + ...","option_d":"1 + x² \/ 2! + x⁴ \/ 4! + ...","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction 1 \/ (1 - x) admet le développement en série entière 1 + x + x² + x³ + ... pour |x| \u003C 1, obtenu par la formule de la série géométrique.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"1 + x + x² + x³ + ...\", \"b\": \"1 - x + x² - x³ + ...\", \"c\": \"x","_debug_options_count":4},{"id":1316,"question":"Pour la série entière Σ (xⁿ \/ n), quel est le rayon de convergence ?","option_a":"R = 0","option_b":"R = 1","option_c":"R = 2","option_d":"R = ∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En appliquant le critère de d'Alembert, on trouve lim |aₙ₊₁ \/ aₙ| = lim n \/ (n+1) = 1, donc le rayon de convergence est R = 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\": \"R = 0\", \"b\": \"R = 1\", \"c\": \"R = 2\", \"d\": \"R = ∞\"}}","_debug_options_count":4}]