• Support PF! Buy your school textbooks, materials and every day products Here!

Recursive sequence - show it does not converge to zero

  • #1
Hi guys,

I'm new here at this forum, but I don't understand this problem.

Let a1 be a positive real number. Define a sequence an recursively by a(n+1) = (an)^2 - 1. Show that an does not converge to zero.

(Is there a1 such that the sequence an converges to some non-zero value?)

I'm not sure if this requires something with epsilon and delta.
 

Answers and Replies

  • #2
22,097
3,282
Can you prove that the limit of the sequence must be a fixed point of [itex]f(x)=x^2-1[/itex]? That is, it satisfies [itex]x=x^2-1[/itex].
 
  • #3
Well I have this:

an = (an)^2 - 1
0 = (an)^2 - an - 1
an = 1 +/- sqrt (5)/2

I'm not sure what this tells me though
 
  • #4
22,097
3,282
See my post 2. Prove that [itex]x=\lim_n a_n[/itex]satisfies [itex]x=x^2-1[/itex].
 
  • #5
a1 = real number > 0

a2 = (a1)^2 - 1 = (a1+1)(a1 - 1)

a3 = (a2)^2 - 1 = (a2 + 1)(a2-1)

a(n+1) = (an)^2 - 1 = (an+1)(an-1). This can only converge to 0 if and only if an equals either +1 or -1. However, that seems contradictory, doesn't it? How can something converge to 0 if it must converge to 1 or -1?
 
Last edited:
  • #6
a1 = real number > 0

a2 = (a1)^2 - 1 = (a1+1)(a1 - 1)

a3 = (a2)^2 - 1 = (a2 + 1)(a2-1)

a(n+1) = (an)^2 - 1 = (an+1)(an-1). This can only converge to 0 if and only if an equals either +1 or -1. However, that seems contradictory, doesn't it? How can something converge to 0 if it must converge to 1 or -1?
 
  • #7
Here is what I am thinking:

An+1=an^2-1
L=L^2-1
0=L^2-L-1
L=1+\- sqrt 5/2 if An converges

Therefore if it does converge it does not
converge to zero

Is that enough?
 

Related Threads on Recursive sequence - show it does not converge to zero

Replies
4
Views
417
  • Last Post
Replies
5
Views
9K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
2
Views
5K
  • Last Post
Replies
13
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
3
Views
305
Replies
8
Views
1K
  • Last Post
Replies
4
Views
2K
Replies
3
Views
3K
Top