Quiz — c-SeriesEntieres.pdf

Question 1 sur 10 20:00

[{"id":26364,"question":"Quel est le rayon de convergence de la série entière Σ (xⁿ \/ n!) ?","option_a":"R = 0","option_b":"R = 1","option_c":"R = +∞","option_d":"R dépend de x","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le rayon de convergence est +∞ car la série converge pour tout x réel (c'est la série exponentielle).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"R = 0\", \"b\": \"R = 1\", \"c\": \"R = +∞\", \"d\": \"R dépend de x\"}}","_debug_options_count":4},{"id":26365,"question":"La série entière Σ n² xⁿ converge-t-elle pour x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour x = 1, la série devient Σ n², qui diverge (terme général non nul et croissant).","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":26366,"question":"Quel critère permet de déterminer le rayon de convergence d'une série entière Σ aₙ xⁿ ?","option_a":"Critère de Cauchy","option_b":"Critère de Leibniz","option_c":"Critère de d'Alembert","option_d":"Critère de Raabe-Duhamel","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le critère de d'Alembert (lim |aₙ₊₁ \/ aₙ|) est couramment utilisé pour les séries entières.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Critère de Cauchy\", \"b\": \"Critère de Leibniz\", \"c\": \"Critère d","_debug_options_count":4},{"id":26367,"question":"La série entière Σ (xⁿ \/ n) a pour rayon de convergence R = 1. Que vaut-elle pour x = -1 ?","option_a":"Elle diverge","option_b":"Elle converge vers ln(2)","option_c":"Elle converge vers -ln(2)","option_d":"Elle converge vers 0","option_e":"","option_f":"","bonne_reponse":"c","explication":"Pour x = -1, la série devient Σ (-1)ⁿ \/ n, qui converge vers -ln(2) (série alternée de Leibniz).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Elle diverge\", \"b\": \"Elle converge vers ln(2)\", \"c\": \"Elle conver","_debug_options_count":4},{"id":26368,"question":"Une série entière peut-elle avoir un intervalle de convergence réduit à un point ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, par exemple la série Σ n! xⁿ a un rayon de convergence R = 0, donc elle ne converge que pour x = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":26369,"question":"Quel est le développement en série entière de eˣ autour de 0 ?","option_a":"Σ xⁿ \/ n!","option_b":"Σ xⁿ \/ n","option_c":"Σ n xⁿ","option_d":"Σ (-1)ⁿ xⁿ \/ n!","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le développement de eˣ est Σ xⁿ \/ n! pour tout x réel, avec un rayon de convergence infini.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Σ xⁿ \/ n!\", \"b\": \"Σ xⁿ \/ n\", \"c\": \"Σ n xⁿ\", \"d\": \"Σ (-1","_debug_options_count":4},{"id":26370,"question":"La série entière Σ (x^(2n) \/ (2n)!) a pour rayon de convergence R = +∞. Que représente-t-elle ?","option_a":"cos(x)","option_b":"ch(x)","option_c":"sin(x)","option_d":"th(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette série représente le cosinus hyperbolique ch(x) = (eˣ + e⁻ˣ)\/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\": \"cos(x)\", \"b\": \"ch(x)\", \"c\": \"sin(x)\", \"d\": \"th(x)\"}}","_debug_options_count":4},{"id":26371,"question":"Peut-on intégrer terme à terme une série entière sur son intervalle de convergence ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, sous réserve de convergence uniforme, l'intégration terme à terme est possible et donne une nouvelle série entière de même rayon de convergence.","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":26372,"question":"Quel est le rayon de convergence de la série entière Σ (n! \/ (2n)!) xⁿ ?","option_a":"R = 0","option_b":"R = 1\/2","option_c":"R = 2","option_d":"R = +∞","option_e":"","option_f":"","bonne_reponse":"d","explication":"En utilisant le critère de d'Alembert, on trouve lim |aₙ₊₁ \/ aₙ| = 0, donc R = +∞.","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 = 1\/2\", \"c\": \"R = 2\", \"d\": \"R = +∞\"}}","_debug_options_count":4},{"id":26373,"question":"La série entière Σ (xⁿ \/ √n) converge-t-elle pour x = -1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour x = -1, la série devient Σ (-1)ⁿ \/ √n, qui converge (série alternée de Leibniz avec terme général décroissant en 1\/√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}]