- #1
al-mahed
- 262
- 0
Hi,
I would like some help in this problem:
given the fibonacci sequence 1,2,3,5,8,13...
we know that a_n/a_(n-1) --> phi when n --> [tex]\infty[/tex]
I have tried a new proof in these terms:
My goal is to prove that when n --> [tex]\infty[/tex], a_n/a_(n-1) = a_(n+1)/a_n in the limit
I have notice that a_n^2= a_(n-1)*a_(n+1) [tex]\pm[/tex] 1 for some values
Dividing both sides of the equation by a_n*a_(n-1)
[tex]\frac{a_n}{a_n_-_1}[/tex] = [tex]\frac{a_n_+_1}{a_n}[/tex][tex]\pm[/tex] [tex]\frac{1}{a_n*a_n_-_1}[/tex]
I want to find the limits, because the second term in the right side of the equation --> 0 when n --> [tex]\infty[/tex]
But I don't know if the equation above is valid for all n, so I'll prove this by induction as follows:
supose that the expression above is valid for some n
by definition a_(n+1)=a_n + a_(n-1)
a_n^2= a_(n-1)*[a_n + a_(n-1)] [tex]\pm[/tex] 1 ==> a_n^2 - a_(n-1)^2= a_(n-1)*a_n [tex]\pm[/tex] 1 ==>
==> [a_n + a_(n-1)]*[a_n - a_(n-1)] = a_(n-1)*a_n [tex]\pm[/tex] 1
by definition a_n - a_(n-1)= a_(n-2) ==>
==> [a_n - a_(n-1)]* a_(n-2) = a_(n-1)*a_n [tex]\pm[/tex] 1
by definition a_n = a_(n-1) + a_(n-2) ==>
==> [a_n - a_(n-1)]* a_(n-2) = a_(n-1)*[a_(n-1) + a_(n-2)] [tex]\pm[/tex] 1 ==>
==> a_(n-2)*a_n + a_(n-1)*a_(n-2) = a_(n-1)^2 + a_(n-1)*a_(n-2) [tex]\pm[/tex] 1 ==>
==> a_(n-1)^2 = a_(n-2)*a_n [tex]\pm[/tex] 1
compare the two expressions
a_n^2= a_(n-1)*a_(n+1) [tex]\pm[/tex] 1
a_(n-1)^2 = a_(n-2)*a_n [tex]\pm[/tex] 1
most generally we have
[tex]\{a}{_i}{^2}[/tex]} = [tex]\{a_i_-_1*a_i_+_1}[/tex]} [tex]\pm[/tex] 1, with i=2,3,4,5,6...,n
proving by induction (sorry about the english and notation)
Hence,
Lim[tex]\{_n_-_>_i_n_f_i_n_i_t_y}[/tex][tex]\frac{a_n}{a_n_-_1}[/tex]} = Lim[tex]\{_n_-_>_i_n_f_i_n_i_t_y}[/tex][tex]\frac{a_n_+_1}{a_n}[/tex][tex]\pm[/tex] [tex]\frac{1}{a_n*a_n_-_1}[/tex]}
Lim[tex]\{_n_-_>_i_n_f_i_n_i_t_y}[/tex][tex]\pm[/tex] [tex]\frac{1}{a_n*a_n_-_1}[/tex]} = 0
in the limit [tex]\frac{a_n}{a_n_-_1}[/tex] = [tex]\frac{a_n_+_1}{a_n}[/tex]
Why this do not prove the convergence?
I would like some help in this problem:
given the fibonacci sequence 1,2,3,5,8,13...
we know that a_n/a_(n-1) --> phi when n --> [tex]\infty[/tex]
I have tried a new proof in these terms:
My goal is to prove that when n --> [tex]\infty[/tex], a_n/a_(n-1) = a_(n+1)/a_n in the limit
I have notice that a_n^2= a_(n-1)*a_(n+1) [tex]\pm[/tex] 1 for some values
Dividing both sides of the equation by a_n*a_(n-1)
[tex]\frac{a_n}{a_n_-_1}[/tex] = [tex]\frac{a_n_+_1}{a_n}[/tex][tex]\pm[/tex] [tex]\frac{1}{a_n*a_n_-_1}[/tex]
I want to find the limits, because the second term in the right side of the equation --> 0 when n --> [tex]\infty[/tex]
But I don't know if the equation above is valid for all n, so I'll prove this by induction as follows:
supose that the expression above is valid for some n
by definition a_(n+1)=a_n + a_(n-1)
a_n^2= a_(n-1)*[a_n + a_(n-1)] [tex]\pm[/tex] 1 ==> a_n^2 - a_(n-1)^2= a_(n-1)*a_n [tex]\pm[/tex] 1 ==>
==> [a_n + a_(n-1)]*[a_n - a_(n-1)] = a_(n-1)*a_n [tex]\pm[/tex] 1
by definition a_n - a_(n-1)= a_(n-2) ==>
==> [a_n - a_(n-1)]* a_(n-2) = a_(n-1)*a_n [tex]\pm[/tex] 1
by definition a_n = a_(n-1) + a_(n-2) ==>
==> [a_n - a_(n-1)]* a_(n-2) = a_(n-1)*[a_(n-1) + a_(n-2)] [tex]\pm[/tex] 1 ==>
==> a_(n-2)*a_n + a_(n-1)*a_(n-2) = a_(n-1)^2 + a_(n-1)*a_(n-2) [tex]\pm[/tex] 1 ==>
==> a_(n-1)^2 = a_(n-2)*a_n [tex]\pm[/tex] 1
compare the two expressions
a_n^2= a_(n-1)*a_(n+1) [tex]\pm[/tex] 1
a_(n-1)^2 = a_(n-2)*a_n [tex]\pm[/tex] 1
most generally we have
[tex]\{a}{_i}{^2}[/tex]} = [tex]\{a_i_-_1*a_i_+_1}[/tex]} [tex]\pm[/tex] 1, with i=2,3,4,5,6...,n
proving by induction (sorry about the english and notation)
Hence,
Lim[tex]\{_n_-_>_i_n_f_i_n_i_t_y}[/tex][tex]\frac{a_n}{a_n_-_1}[/tex]} = Lim[tex]\{_n_-_>_i_n_f_i_n_i_t_y}[/tex][tex]\frac{a_n_+_1}{a_n}[/tex][tex]\pm[/tex] [tex]\frac{1}{a_n*a_n_-_1}[/tex]}
Lim[tex]\{_n_-_>_i_n_f_i_n_i_t_y}[/tex][tex]\pm[/tex] [tex]\frac{1}{a_n*a_n_-_1}[/tex]} = 0
in the limit [tex]\frac{a_n}{a_n_-_1}[/tex] = [tex]\frac{a_n_+_1}{a_n}[/tex]
Why this do not prove the convergence?