Maîtrisez les suites réelles : limites, convergence et récurrence

Question 1 sur 10 10:00

[{"id":29788,"question":"Quelle est la limite de la suite u_n = 1\/n lorsque n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"+∞","option_d":"−∞","option_e":"","option_f":"","bonne_reponse":"A","explication":"La limite de 1\/n quand n tend vers l'infini est 0, car le dénominateur devient de plus en plus grand.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29789,"question":"Une suite croissante est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une suite croissante peut diverger vers +∞ (exemple : u_n = n). Elle n'est convergente que si elle est majorée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29790,"question":"Soit la suite définie par u_0 = 2 et u_{n+1} = 2u_n - 1. Quelle est la nature de cette suite ?","option_a":"Arithmétique","option_b":"Géométrique","option_c":"Constante","option_d":"Ni l'une ni l'autre","option_e":"","option_f":"","bonne_reponse":"B","explication":"Cette suite est géométrique de raison 2, car u_{n+1} = 2u_n.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29791,"question":"La suite u_n = (-1)^n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Cette suite n'a pas de limite car elle oscille entre -1 et 1 sans se stabiliser.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29792,"question":"Quelle est la limite de la suite u_n = n\/(n+1) lorsque n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"1\/2","option_d":"+∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"En divisant numérateur et dénominateur par n, on obtient u_n = 1\/(1+1\/n), dont la limite est 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29793,"question":"Pour démontrer qu’une suite est bornée, il suffit de montrer qu’elle est majorée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une suite bornée doit être à la fois majorée et minorée. Montrer seulement qu'elle est majorée ne suffit pas.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29794,"question":"Soit la suite u_n = 3^n. Quelle est sa limite lorsque n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"3","option_d":"+∞","option_e":"","option_f":"","bonne_reponse":"D","explication":"La suite 3^n tend vers +∞ car la base est supérieure à 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29795,"question":"Une suite décroissante et minorée est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une suite décroissante et minorée converge vers sa borne inférieure (théorème de la limite monotone).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29796,"question":"Quelle est la limite de la suite u_n = (2n+1)\/(n-3) lorsque n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"2","option_d":"+∞","option_e":"","option_f":"","bonne_reponse":"C","explication":"En divisant numérateur et dénominateur par n, on obtient u_n = (2+1\/n)\/(1-3\/n), dont la limite est 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29797,"question":"La suite u_n = n^2 - 5n + 6 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Cette suite est un polynôme de degré 2, donc elle diverge vers +∞ lorsque n tend vers l'infini.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]