Find lt n->(infinity) f(n)

  • #1

Homework Statement


Given:
[tex]f(n+1)=\frac{1}{2}(f(n)+\frac{9}{f(n)}[/tex]

Find lt n->(infinity) f(n)

Homework Equations




The Attempt at a Solution



No idea.

[tex]f(n+1)-f(n)=0.5(\frac{9}{f(n)} -f(n))[/tex]
Replacing n by 1/n, dividing by 1/n and taking limit 1/n->0 which becomes something similar to the f'(n) except 1 should be h. Even still, you cant apply L'hospitals here. Im totally lost.
 

Answers and Replies

  • #2
AKG
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In your equation f(n+1) = 0.5(f(n) - 9/f(n)), the left side equals the right side, therefore the limit as n goes to infinity of the right side is the limit as n goes to infinity of the right side. Using the above fact, let L stand for the limit of f(n) as n goes to infinity, and solve for L.
 
  • #3
HallsofIvy
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Your orginal equation was
[tex]f(n+1)= \frac{1}{2}a(f(n)+ 9/f(n)[/tex]
I think your subtracting f(n) from both sides confused AKG (not to mention me)!

The sequence {f(n+1)} is exactly the same as {f(n)} but with the index changed. If L= lim f(n) then L= lim f(n+1) also. AKG's suggestion is that you take the limit, as n goes to infinity, on both sides of the equation and so get a single equation involving L. Solve that equation.
 
  • #4
So, the equation should be something like L=0.5(L+9/L)?
Solving this gives you L=+3 and L=-3. What should my answer be?
 
  • #5
AKG
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Well if you've posted the question exactly as it's given to you, then both 3 and -3 are possible. Observe that if f is a sequence that satisfies the given relation, then so is -f, so if the limit of f is 3, then the limit of -f is -3.
 
  • #6
HallsofIvy
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In other words, whether the limit is 3 or -3 depends upon f(0).

If, for example, you know that f(0)= 1, then f(1)= (1/2)(1+ 9/1)= 5, f(2)= (1/2)(5+ 9/5)= 6.8/2= 3.4, f(3)= (1/2)(3.4+ 9/3.4)= 3.0234.., converging to 3.

If, instead, f(0)= -1, then f(1)= (1/2)(-1+9/-1)= -5, etc. converging to -3.
 

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