{"id":4192,"date":"2015-03-10T06:47:59","date_gmt":"2015-03-10T04:47:59","guid":{"rendered":"https:\/\/kamerpower.com\/?p=4192"},"modified":"2025-06-29T03:39:54","modified_gmt":"2025-06-29T02:39:54","slug":"epreuve-de-1ere-composition-de-mathematiques-issea-yaounde-2014-ensea-abidjan-2014","status":"publish","type":"post","link":"https:\/\/kamerpower.com\/fr\/epreuve-de-1ere-composition-de-mathematiques-issea-yaounde-2014-ensea-abidjan-2014\/","title":{"rendered":"\u00c9preuve de 1\u00e8re composition de Math\u00e9matiques ISSEA Yaounde 2014 \/ ENSEA Abidjan 2014 – Concours Ing\u00e9nieurs statisticiens \u00e9conomistes – ISE"},"content":{"rendered":"

\u00c9COLE NATIONALE SUP\u00c9RIEURE<\/span> — INSTITUT SOUS-R\u00c9GIONAL DE STATISTIQUE<\/span><\/span>
\nDE STATISTIQUE ET D\u2019ECONOMIE<\/span> — ET D\u2019\u00c9CONOMIE APPLIQU\u00c9E<\/span><\/span>
\nENSEA – ABIDJAN<\/span> — ISSEA – YAOUND\u00c9<\/span><\/span><\/p>\n

\u00c9COLE NATIONALE DE LA STATISTIQUE<\/span>
\nET DE L’ANALYSE \u00c9CONOMIQUE<\/span>
\nENSAE – S\u00c9N\u00c9GAL<\/span>
\nAVRIL 2014<\/span><\/p>\n

CONCOURS ING\u00c9NIEURS STATISTICIENS \u00c9CONOMISTES<\/a><\/span><\/p>\n

ISE Option \u00c9conomie<\/span>
\n1\u00e8re\u00a0COMPOSITION DE MATH\u00c9MATIQUES<\/span>
\n(Dur\u00e9e de l\u2019\u00e9preuve : 4\u00a0heures)
\nKA<\/span>M<\/span>ER<\/span>POWER<\/span>.COM<\/span><\/span><\/p>\n

\u00c9preuve de 1\u00e8re composition de Math\u00e9matiques ISSEA Yaounde 2014 \/ ENSEA Abidjan 2014 – Concours Ing\u00e9nieurs statisticiens \u00e9conomistes – ISE 2014<\/strong><\/span><\/p>\n

L\u2019\u00e9preuve comporte un seul probl\u00e9me, compos\u00e9 de quatre parties non ind\u00e9pendantes not\u00e9es\u00a0de A \u00e0\u00a0D.<\/span><\/p>\n

Probl\u00e8me\u00a0<\/span><\/h2>\n

Le symbole Ln d\u00e9signe le logarithme n\u00e9p\u00e9rien de base e, e = 2,718.<\/span><\/p>\n

Partie A<\/span><\/strong><\/p>\n

On consid\u00e9re la fonction num\u00e9rique h de la variable r\u00e9elle x, strictement positive, d\u00e9finie par:<\/span><\/p>\n

h(x) = \\frac{1}{2}(x – \u00a0\\frac{1}{x})\u00a0 – Lnx<\/p>\n

Etudier tr\u00e9s pr\u00e9cis\u00e9ment les variations de h.<\/span>
\n On \u00e9tudiera en particulier le\u00a0signe et les variations de h^{‘}\u00a0et \u00a0h^{”} pour \u00e9tablir le tableau complet<\/span>
\n des variations de h ; on n\u2019oubliera pas les\u00a0points caract\u00e9ristiques, leurs tangentes, les limites et\u00a0<\/span>les asymptotes \u00e9ventuelles de h, etc.<\/span><\/p>\n

Partie B<\/span><\/strong><\/p>\n

1)<\/span> Soient les deux fonctions a(t) et b(t) de la variable r\u00e9elle t,\u00a0t\\in J = ] – 1, + \u221e\u00a0[, d\u00e9finies par:<\/span><\/p>\n

a(t) = \\frac{1}{t+1}\u00a0et \u00a0 b(t) = Ln(1+t)<\/span><\/p>\n

Donner les d\u00e9veloppements limit\u00e9s a l\u2019ordre 3 de a(t) et b(t) au voisinage de 0.<\/span><\/p>\n

2)<\/span> Montrer que \u00a0 \u00a0\u00a0 , pour t\\in J<\/strong><\/span><\/p>\n

3)<\/span> On consid\u00e8re la fonction num\u00e9rique f de la variable r\u00e9elle\u00a0t\\in J<\/strong>\u00a0= ] – 1, + \u221e\u00a0[, d\u00e9finie par<\/span><\/p>\n

et \u00a0 \u00a0 f(0) = 1<\/p>\n

3a<\/span> – Quel est le\u00a0signe de f ?<\/span>
\n3b<\/span> – Montrer que f est continue en 0.<\/span>
\n4)<\/span> Calculer f\u00a0‘(t). Montrer que f est d\u00e9rivable en 0.<\/span><\/p>\n

\u00a0Partie\u00a0C<\/span><\/strong><\/p>\n

1)<\/span> On consid\u00e8re la fonction num\u00e9rique g de la variable r\u00e9elle t,\u00a0t\\in J<\/strong>\u00a0\u00a0= ]-1, + \u221e\u00a0[, d\u00e9finie par<\/span><\/p>\n

\u00c9tablir un lien entre g et h (h introduite \u00e0 la Partie A).<\/span><\/p>\n

2)<\/span> Montrer que la deriv\u00e9e f ‘(t) peut \u00e9tre mise sous la forme \u00a0\u00a0, pour t\u00a0\\neq<\/strong> \u00a00.<\/span><\/p>\n

3)<\/span> D\u00e9montrer que f ‘\u00a0est continue pour tout\u00a0t\\in J<\/strong>\u00a0.<\/span>
\n 4)<\/span> A partir des tableaux de variations de g et f, montrer que f(t)\u00a0\\geq \u00a01 pour tout\u00a0t\\in J<\/strong>\u00a0.<\/span><\/p>\n

5)<\/span> Montrer que \u00a0\u00a0\u00a0pour\u00a0t\\in<\/strong> J\u00a0.<\/span><\/p>\n

\u00a0Partie D<\/span><\/strong><\/p>\n

Soit une suite {u_{{n}} }, \u00e0 termes positifs, n entier > 0\u00a0; on d\u00e9finit la suite {U_{{n}}<\/strong>\u00a0} par:<\/span><\/p>\n

U_{{n}} = \\sum_{k=1}^{n} u_{{k}}<\/p>\n

1)<\/span> Montrer qu\u2019une condition n\u00e9cessaire pour que la suite {U_{{n}}<\/strong>} \u00a0admette une limite finie U est\u00a0<\/span>que\u00a0u_{{n}}\\to0<\/strong>\u00a0\u00a0quand n\\to+<\/strong> \u221e.<\/span><\/p>\n

2)<\/span> Que peut-on dire du comportement de la suite {U_{{n}}<\/strong> } si u_{{n}}<\/strong> ne tend pas vers 0\u00a0\u00a0quand<\/span>
\n n \u00a0+\u221e ?<\/span><\/p>\n

3)<\/span> Pour x r\u00e9el > 0, n entier > 0, on d\u00e9finit la suite de fonctions {u_{{a}}(x)<\/strong>} de terme g\u00e9n\u00e9ral<\/span><\/p>\n

Montrer que, pour x > 0, le ratio \u00a0est \u00e9gal \u00e0 e.x. o\u00f9 f a \u00e9t\u00e9 d\u00e9finie en B3.<\/span><\/p>\n

4)<\/span> Dans cette question, on suppose que\u00a0x\\geq \\frac{1}{e}<\/strong><\/span><\/p>\n

4a<\/span> – Montrer que la suite de terme g\u00e9n\u00e9ral {u_{{n}}(x) }, n entier > 0, est une suite croissante.<\/span><\/p>\n

4b<\/span> – En d\u00e9duire alors la nature de la suite associ\u00e9e {U_{{n}}(x)<\/strong>} o\u00f9\u00a0U_{{n}}(x)= \\sum_{k=1}^{n} u_{{k}}(x)<\/strong>.<\/span><\/p>\n

5)<\/span> Dans cette question, on suppose que ex < l.<\/span>
\nSoit q un nombre r\u00e9el tel que ex < q < 1.<\/span><\/p>\n

5a<\/span> – Montrer qu\u2019il existe un entier N tel que, pour tout\u00a0n\\geq N<\/strong>, u_{{n+1}}(x)\/u_{{n}}(x)\\leq q<\/strong>.<\/span><\/p>\n

5b<\/span> – En d\u00e9duire alors que, pour tout n\u00a0\\geq<\/strong> \u00a0N,\u00a0u_{{n}}(x)\\leq q ^{n-N}u_{{N}}(x)<\/strong>.<\/span><\/p>\n

5c<\/span> – Quelle est la nature de la suite\u00a0{U_{{n}}(x)<\/strong>} ?<\/span><\/p>\n

6)<\/span> Pour n entier > 0\u00a0, on d\u00e9finit les deux suites (v_{{n}})<\/strong> et (w_{{n}}<\/strong>) par :<\/span><\/p>\n

et<\/p>\n

Quelles sont les limites de ces deux suites ?<\/span><\/p>\n

7)<\/span> On \u00e9tudie la suite\u00a0u_{{n}}(\\frac{1}{e})<\/strong>.<\/span><\/p>\n

7a<\/span> – Donner l\u2019expression de u_{{n}}(\\frac{1}{e})<\/strong>.<\/span><\/p>\n

7b<\/span>\u00a0– Pour tout entier\u00a0k\\geq 1<\/strong>, calculer le rapport <\/span><\/p>\n

7c<\/span> – Montrer, en utilisant la question 5<\/span> de la partie C<\/span>, que Ln R(k, c) \u00a0est major\u00e9 par\u00a0<\/span><\/p>\n

7d<\/span> – En d\u00e9duire que, pour n entier > 1 :<\/span><\/p>\n

7e<\/span> – Montrer que la suite \u00a0est major\u00e9e.<\/span><\/p>\n

8)<\/span> Soit\u00a0la suite\u00a0\u00a0, n entier > 0.<\/span><\/p>\n

8a<\/span> – Donner une relation entre\u00a0z_{{n}}<\/strong>\u00a0et \u00a0<\/span><\/p>\n

8b<\/span> – Quelle est la limite de la suite\u00a0z_{{n}}<\/strong>\u00a0quand n tend vers +\u00a0\u221e ?<\/span><\/p>\n

\n

\u00c9preuve de 1\u00e8re composition de Math\u00e9matiques ISSEA Yaounde 2014 \/ ENSEA Abidjan 2014 – Concours Ing\u00e9nieurs statisticiens \u00e9conomistes – ISE 2014<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

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