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Topic: Analysys with f(f(a_n))
Replies: 4   Last Post: Jun 6, 2013 5:33 AM

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William Elliot

Posts: 1,608
Registered: 1/8/12
Re: Analysys with f(f(a_n))
Posted: Jun 6, 2013 5:33 AM
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On Wed, 5 Jun 2013, mina_world@hanmail.net wrote:

> f(x) = x + 2 x <= 0
> -x/2 x > 0


> a_1 = 1
> a_(n+1) = f(f(a_n)) (n >= 1)


> Find the lim{n->oo} a_n

If the limit a exists, would a = ff(a)?

ff(x) = f(x+2) = x + 4 x <= -2
ff(x) = f(x+2) = -x/2 - 1 -2 < x <= 0
ff(x) = f(-x/2) = -x/2 + 2 0 < x

a = -a/2 - 1; 2a = -a - 2; 3a = -2; no, not a <= -2
a = -a/2 + 2; 2a = -a + 4; 3a = 4; yes, 0 < a

0 < ff(x) < 2, if 0 < x < 2.
In addition ff is continuous over (0,2), so we can use a = ff(a).

Let g = ff = f o f. a2 = g(a1) = g(1) = 3/2
a2 = g(a1) = g(3/2) = 5/4
a3 = g(a2) = g(5/4) = 11/8

Does a pattern emerge?




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