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Convergencia de Secuencias

Module by: Richard Baraniuk Translated By Fara Meza, Erika JacksonBased on: Convergence of Sequences by Richard Baraniuk

Summary: Este módulo presentra una introducción a la convergencia y se enfocara en la secuencia, su comportamiento y sus aproximaciones al infinito.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

¿Qué es una Secuencia?

Definition 1: secuencia
Una secuencia es una función gn gn definida en los enteros positivos 'nn'. También denotamos una secuencia por: gn |n=1∞ n 1 gn

Ejemplo

Una secuencia de números reales: gn=1n gn 1 n

Ejemplo

Una secuencia vector: gn=sinnπ2cosnπ2 gn n 2 n 2

Ejemplo

Una secuencia de una función: g n t=1if0≤t<1n0otherwise g n t 1 0 t 1 n 0

nota:

Una función puede ser pensada como un vector de dimensión infinito donde por cada valor de 'tt' tenemos una dimensión.

Convergencia de Secuencias Reales

Definition 2: límite
Una secuencia gn |n=1∞ n 1 gn converge a un límite g∈ℝ g si para todo ε>0 ε 0 existe un entero N N tal que ∀i,i≥N:|gi−g|<ε i i N gi g ε Usualmente denotamos un límite escribiendo limi→∞gi=g i gi g ó g i →g g i g
La definición anterior significa que no importa que tan pequeño hagamos εε, excepto para un número finito de g i g i 's, todos los puntos de la secuencia estan dentro de una distancia εε de gg.

Ejemplo 1

Dada las siguiente secuencia de convergencia :

gn=1n gn 1 n (1)
Intuitivamente podemos asumir el siguiente límite: limn→∞gn=0 n gn 0 . Ahora provemos esto rigurosamente, digamos que dado el número real ε>0 ε 0 . Escojamos N=⌈1ε⌉ N 1 ε , donde ⌈x⌉ x denota el menor entero mayor que xx. Entonces para n≥N n N tenemos |gn−0|=1n≤1N<ε gn 0 1 n 1 N ε Donde, limn→∞gn=0 n gn 0

Ejemplo 2

Ahora veamos la siguiente secuencia no-convergente gn= 1ifn=par-1ifn=impar gn 1 n par -1 n impar Esta secuencia oscila entre 1 y -1, asi que nunca converge.

Problemas

Para practicar, diga cuales de las siguientes secuencias convergen y de el limite si este existe.

  1. gn=n gn n
  2. gn= 1nifn=par-1nifn=impar gn 1 n n par -1 n n impar
  3. gn= 1nifn≠potencia de 101otherwise gn 1 n n potencia de 10 1
  4. gn=nifn<1051nifn≥105 gn n n 10 5 1 n n 10 5
  5. gn=sinπn gn n
  6. gn=ⅈn gn n

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