auxiliar10_MA1001_2012.pdf - Universidad de Chile Facultad De Ciencias F\u00b4\u0131sicas y Matem\u00b4aticas Prof Ra\u00b4 ul Uribe Prof Aux Braulio S\u00b4anchez Ib\u00b4an

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Universidad de Chile Facultad De Ciencias F´ ısicas y Matem´ aticas Prof.: Ra´ul Uribe Prof. Aux.: Braulio S´ anchez Ib´ a˜nez Introducci´on al C´ alculo Clase Auxiliar 10 - Sucesiones II 31 de mayo de 2012 Problema 1 [Sucesiones recursivas].- (a) Considere la sucesi´ on definida mediante la recurrencia a n +1 = a n 2 ( 1 + a 2 n - 1 ) , con a 0 , a 1 (0 , 1) (i) Demuestre que n N , a n (0 , 1) (ii) Muestre que ( a n ) es convergente. (iii) Calcule el l´ ımite de la sucesi´on. (b) Considere los reales 0 < a < b . Se definen las sucesiones ( x n ), ( y n ) mediante las siguientes recurrencias x 1 = a , x n +1 = x n y n y 1 = b , y n +1 = x n + y n 2 Demuestre que ambas sucesiones convergen, que poseen el mismo l´ ımite l y que ab < l < a + b 2 (c) Considere las sucesiones reales definidas por x n = n X k =0 1 k ! y n = x n + 1 n · n ! (i) Pruebe que ( x n ) e ( y n ) son sucesiones mon´ otonas. (ii) Demuestre que ambas sucesiones convergen, que lo hacen al mismo l´ ımite l , que n N * , x n < l < y n y que 2 , 5 < l < 2 , 75.

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