The Fibonacci sequence can be defined at $F_{n+1} = F_n + F_{n-1}$ for $n\ge0$ and with $F_0 = 0$ and $F_1 = 1$. It can be shown that the ratio between successive terms converges to

$$ \lim_{n\rightarrow\infty}\frac{F_{n+1}}{F_n} = \varphi = \frac{1+\sqrt{5}}{2}\approx 1.618$$

It seems harder to show exactly how quickly this convergence happens. Typically the rate of convergence would be computed as

$$ \lim_{n\rightarrow\infty} \frac{|\frac{F_{n+1}}{F_n}-\varphi|}{|\frac{F_n}{F_{n-1}}-\varphi|} = \mu$$

but this is hard to compute and for me has been going to either 0, 1, or $\varphi$ on paper. By computer the ratio of successive terms in the Fibonacci seqence shows a linear convergence, i.e., $\mu \in [0,1]$, of $1/\varphi^2 \approx .382$.

Can anyone show this analytically?


You get that $\frac{F_{n+1}}{F_n}$ alternates around $φ$ with $$ \frac{F_{n+1}}{F_n}-\frac{F_n}{F_{n-1}}=\frac{(-1)^{n-1}}{F_nF_{n-1}} $$ so that indeed the distance to $φ$ falls like $φ^{-2n}$.

If you want to be more exact, you know that $F_n=aφ^n+b(-φ)^{-n}$, $a\ne 0$, so that $$ \frac{F_{n+1}}{F_n}=φ\frac{a+b(-φ^2)^{-n-1}}{a+b(-φ^2)^{-n}} \implies \frac{F_{n+1}}{F_n}-φ=φ\frac{b(-φ^2)^{-n-1}(1-φ^2)}{a+b(-φ^2)^{-n}} $$ and thus the rate of linear convergence is $$ \frac{\frac{F_{n+1}}{F_n}-φ}{\frac{F_n}{F_{n-1}}-φ}=-φ^{-2}\frac{a+b(-φ^2)^{-n+1}}{a+b(-φ^2)^{-n}} $$ which converges to $-φ^{-2}$.

  • $\begingroup$ It looks like you are jumping to the limit of the absolute error for the sequence, instead of evaluating the limit I gave above? Also, I do not follow how you went from the left to the right side. $\endgroup$ – guitarphish Jul 6 '19 at 16:58
  • $\begingroup$ Thank you for helping, by the way! The limit is the definition of the rate of convergence (en.wikipedia.org/wiki/Rate_of_convergence). If you are calculating the limit of the absolute error, which is what it looks like, that would be roughly the same thing (but it might not work for non-linear rates of convergence, not sure). So I am understanding the left side of your expression, but I am unable to find a path to the right side. $\endgroup$ – guitarphish Jul 6 '19 at 17:09
  • $\begingroup$ In your "more exact" formula involving $a$ and $b$, you probably want $-1/\varphi$, which is $\frac{1-\sqrt5}2$, rather than $-\varphi$. $\endgroup$ – Andreas Blass Jul 6 '19 at 17:15
  • $\begingroup$ @AndreasBlass : That's why the exponent is negative. As in $-1/φ=(-φ)^{-1}$. $\endgroup$ – Lutz Lehmann Jul 6 '19 at 17:38
  • $\begingroup$ I've worked it out to confirm that this works, the algebra is thick in this one but I ended up getting the anticipated result! Thank you again! $\endgroup$ – guitarphish Jul 6 '19 at 18:11

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