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Calculus/Fibonacci convergence

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Expert: - 12/10/2008


Question
I was given a problem by my professor to be graded as a quiz.  It is to find the exact expression for the ratio of successive Fibonacci terms.  From my internet searches, I know this is phi (1 +/- sqrt 5)/2.  After talking to him I know I'm on the right track, but can't finish it.  I started with F(n+1) = F(n) + F(n-1).  Then divided by F(n) and took the limit as n approaches infinity.  What I got was:
lim (n->infinity) F(n+1)/F(n) = 1 + lim (n->infinity) F(n-1)/f(n).  Where I'm stuck is how to evaluate that limit on the right.  My professor told me to think what that means in words and what happens to that expression as n gets really large.  I can't for the life of me figure it out.  Also we're not supposed to know what the value is until we find it this way.  Can you help nudge me in the right direction?

Answer
Note that in this equation
lim (n->∞) F(n+1)/F(n) = 1 + lim (n->∞) F(n-1)/F(n),
we know that F(n+1)=F(n)+F(n-1), so this can be put into the left side, giving lim (n->∞) (F(n)+F(n-1))/F(n).

Factor, note the F(n)/F(n) is 1 and can be moved outside, leaving you with... you'll get it.

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