
Re: Analysys with f(f(a_n))
Posted:
Jun 6, 2013 5:33 AM


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?

