Quiz — probleme serie numérique.pdf

Question 1 sur 10 20:00

[{"id":33390,"question":"Quelle est la nature de la série de terme général 1\/n² ?","option_a":"Convergente","option_b":"Divergente","option_c":"Oscillante","option_d":"Indéterminée","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série de terme général 1\/n² est une série de Riemann avec α=2\u003E1, donc elle converge (critère de Riemann).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Convergente\", \"b\": \"Divergente\", \"c\": \"Oscillante\", \"d\": \"Indéte","_debug_options_count":4},{"id":33391,"question":"Le critère de comparaison peut être appliqué pour comparer deux séries à termes positifs.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le critère de comparaison nécessite que les deux séries soient à termes positifs et que l'une soit majorée ou minorée par l'autre.","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":33392,"question":"Quelle est la limite de la série géométrique de premier terme 1 et de raison 1\/2 ?","option_a":"1","option_b":"2","option_c":"∞","option_d":"0","option_e":"","option_f":"","bonne_reponse":"b","explication":"La somme d'une série géométrique de raison |r|\u003C1 est S = u₁\/(1-r) = 1\/(1-1\/2) = 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\": \"1\", \"b\": \"2\", \"c\": \"∞\", \"d\": \"0\"}}","_debug_options_count":4},{"id":33393,"question":"Le critère de d'Alembert permet de conclure à la convergence d'une série si la limite de |uₙ₊₁\/uₙ| est strictement inférieure à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si lim |uₙ₊₁\/uₙ| = L \u003C 1, alors la série converge absolument (critère de d'Alembert).","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":33394,"question":"Quelle est la nature de la série de terme général (-1)ⁿ\/n ?","option_a":"Convergente","option_b":"Divergente","option_c":"Semi-convergente","option_d":"Indéterminée","option_e":"","option_f":"","bonne_reponse":"c","explication":"Cette série est semi-convergente (converge mais pas absolument) car la série des valeurs absolues est la série harmonique, divergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Convergente\", \"b\": \"Divergente\", \"c\": \"Semi-convergente\", \"d\": \"I","_debug_options_count":4},{"id":33395,"question":"Le critère de Cauchy (ou critère de la racine) s'applique uniquement aux séries à termes positifs.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le critère de Cauchy peut s'appliquer à des séries à termes réels (positifs ou négatifs) en considérant la valeur absolue.","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":33396,"question":"Quelle est la somme de la série de terme général 1\/2ⁿ pour n allant de 0 à ∞ ?","option_a":"1","option_b":"2","option_c":"1\/2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Il s'agit d'une série géométrique de premier terme 1 et de raison 1\/2, donc S = 1\/(1-1\/2) = 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\": \"1\", \"b\": \"2\", \"c\": \"1\/2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":33397,"question":"Si une série converge absolument, alors elle converge.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La convergence absolue implique la convergence simple (théorème de comparaison absolue).","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":33398,"question":"Quelle est la nature de la série de terme général n\/(n+1) ?","option_a":"Convergente","option_b":"Divergente","option_c":"Oscillante","option_d":"Indéterminée","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le terme général n\/(n+1) tend vers 1 ≠ 0, donc la série diverge (critère de divergence du terme général).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Convergente\", \"b\": \"Divergente\", \"c\": \"Oscillante\", \"d\": \"Indéte","_debug_options_count":4},{"id":33399,"question":"Le critère de Raabe-Duhamel permet de conclure à la convergence si lim n(1 - |uₙ₊₁\/uₙ|) \u003E 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si lim n(1 - |uₙ₊₁\/uₙ|) = L \u003E 1, alors la série converge (critère de Raabe-Duhamel).","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}]