Sucesiones 3 - (xn es la sucesin denida como o(xn = 1(1)n 1...

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( x n ) es la sucesi´ on definida como: ( x n ) = 1 + ( - 1) n +1 P.D. Que x n no es convergente. Supongamos que x n es convergente, tenemos que l´ ım x n = a entonces toda subsucesi´ on de ( x n ) converge a a . Sea ( x n k ) = (0 , 0 , 0 , ... ) y ( x n k 0 ) = (2 , 2 , 2 , ... ) subsucesiones de ( x n ): ( x n k ) = (0 , 0 , 0 , ... ): x : N R n 7→ x n = 1 + ( - 1) n +1 α : N N k 7→ 2 k x α : N R k 7→ = x α = x ( α ( k )) = x (2 k ) = 1 + ( - 1) 2 k +1 = 1 + ( - 1) = 0 ( x n k ) = (2 , 2 , 2 , ... ): x : N R n 7→ x n = 1 + ( - 1) n +1 β : N N k 7→ 2 k - 1 x β : N R k 7→ = x β = x ( β ( k )) = x (2 k ) = 1 + ( - 1) 2 k - 1+1 = 1 + ( - 1) 2 k = 2 Los L´ ımites de cada subsucesi´ on son: ım x n k = 0 ım x n k 0 = 2 Por: ımite de una constante es la misma constante: P.D.l´
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  • Winter '09
  • CarlosTrujillo
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