Quiz — série entière.pdf

Question 1 sur 10 20:00

[{"id":67976,"question":"Quelle est la forme générale d'une série entière centrée en 0 ?","option_a":"Σ aₙ xⁿ","option_b":"Σ aₙ (x - x₀)ⁿ","option_c":"Σ n aₙ xⁿ","option_d":"Σ (x - x₀)ⁿ \/ n!","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une série entière centrée en 0 s'écrit Σ aₙ xⁿ, où aₙ sont les coefficients et x la variable.","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 aₙ xⁿ\", \"b\": \"Σ aₙ (x - x₀)ⁿ\", \"c\": \"Σ n aₙ xⁿ","_debug_options_count":4},{"id":67977,"question":"Le critère de d'Alembert permet de déterminer la convergence d'une série.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le critère de d'Alembert (ou critère du rapport) est un outil pour étudier la convergence absolue d'une série.","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":67978,"question":"Pour une série géométrique de raison r, la série converge si :","option_a":"|r| \u003C 1","option_b":"|r| \u003E 1","option_c":"r = 1","option_d":"r = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une série géométrique Σ rⁿ converge si et seulement si la raison r vérifie |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\": \"|r| \u003C 1\", \"b\": \"|r| \u003E 1\", \"c\": \"r = 1\", \"d\": \"r = 0\"}}","_debug_options_count":4},{"id":67979,"question":"Quelle est la somme d'une série géométrique convergente de premier terme a et de raison r ?","option_a":"a \/ (1 - r)","option_b":"a (1 - r)","option_c":"a r","option_d":"a \/ r","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme d'une série géométrique convergente est donnée par S = a \/ (1 - r), où a est le premier terme et r la raison.","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 \/ (1 - r)\", \"b\": \"a (1 - r)\", \"c\": \"a r\", \"d\": \"a \/ r\"}}","_debug_options_count":4},{"id":67980,"question":"Le critère de Cauchy (ou critère de la racine) s'applique à toutes les séries.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le critère de Cauchy s'applique principalement aux séries à termes positifs ou aux séries absolument convergentes.","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":67981,"question":"Quelle est la condition nécessaire pour qu'une série Σ uₙ converge ?","option_a":"lim uₙ = 0","option_b":"lim uₙ ≠ 0","option_c":"uₙ \u003E 0 pour tout n","option_d":"uₙ est croissante","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une condition nécessaire (mais non suffisante) pour la convergence d'une série est que son terme général uₙ tende vers 0 lorsque n tend vers l'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\": \"lim uₙ = 0\", \"b\": \"lim uₙ ≠ 0\", \"c\": \"uₙ \u003E 0 pour tout n\"","_debug_options_count":4},{"id":67982,"question":"Une série entière a toujours un rayon de convergence strictement positif.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le rayon de convergence d'une série entière peut être nul (série convergente uniquement en x = x₀) ou infini (série convergente pour tout x).","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":67983,"question":"Quelle est la dérivée terme à terme d'une série entière Σ aₙ xⁿ ?","option_a":"Σ n aₙ xⁿ⁻¹","option_b":"Σ aₙ xⁿ","option_c":"Σ n² aₙ xⁿ","option_d":"Σ (n+1) aₙ xⁿ","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée terme à terme d'une série entière Σ aₙ xⁿ est Σ n aₙ xⁿ⁻¹, sous réserve 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\": \"Σ n aₙ xⁿ⁻¹\", \"b\": \"Σ aₙ xⁿ\", \"c\": \"Σ n² aₙ xⁿ","_debug_options_count":4},{"id":67984,"question":"Le rayon de convergence d'une série entière est toujours un nombre réel fini.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le rayon de convergence peut être un nombre réel fini, nul ou infini.","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":67985,"question":"Quelle est l'intégrale terme à terme d'une série entière Σ aₙ xⁿ ?","option_a":"Σ (aₙ xⁿ⁺¹) \/ (n+1) + C","option_b":"Σ aₙ xⁿ","option_c":"Σ n aₙ xⁿ","option_d":"Σ (aₙ \/ (n+1)) xⁿ⁺¹","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale terme à terme d'une série entière Σ aₙ xⁿ est Σ (aₙ xⁿ⁺¹) \/ (n+1) + C, sous réserve 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\": \"Σ (aₙ xⁿ⁺¹) \/ (n+1) + C\", \"b\": \"Σ aₙ xⁿ\", \"c\": \"Σ n","_debug_options_count":4}]