any help on getting started would be great. i really don't know where to go with this as i'm useless with limits.
alpha is the golden ration ((1+sqrt5)/2)
f2k etc are part of the fibonacci sequence
prove lim f2k/f2k+1 = 1/alpha
k->infinity
any help on getting started would be great. i really don't know where to go with this as i'm useless with limits.
alpha is the golden ration ((1+sqrt5)/2)
f2k etc are part of the fibonacci sequence
prove lim f2k/f2k+1 = 1/alpha
k->infinity
The Fibonacci sequence can be written as a recurrence relation. Solve that relation for the nth term, and then form the desired ratio and take the limit. Does that give you some ideas?
The Fibonacci's sequence is the solution of the 'recursive relation'...
$\displaystyle \displaystyle y_{n} = y_{n-1} + y_{n-2}\\, y_{0}=0\\, \\y_{1}=1$ (1)
The (1) is linear and homogenous and its general solution is...
$\displaystyle \displaystyle y_{n} = c_{1}\ \varphi^{n} + c_{2}\ (-\varphi)^{-n}$ (2)
... where $\displaystyle \varphi = \frac{1+\sqrt{5}}{2}$ is the 'golden ratio'. Now in (2) is...
$\displaystyle \displaystyle \lim_{n \rightarrow \infty} (-\varphi)^{-n}=0$ (3)
... so that is...
$\displaystyle \displaystyle \lim_{n \rightarrow \infty} \frac{y_{n}}{y_{n-1}} = \varphi$ (4)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$