Continued Fractions Homework: Find Explicit Formula

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In summary, the person is trying to solve a recursive equation that involves Fibonacci numbers, but is having difficulty. They eventually figure out how to solve the equation using the exact method.
  • #1

Homework Statement



Give the formula in terms of tn+1 for the continued fraction:
http://www.math.sunysb.edu/posterproject/www/images/continued-fraction.gif [Broken]
and so on...

Homework Equations


The Attempt at a Solution


I got the recursive formula:tn+1=1+(1/tn), but I need the explicit formula
 
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  • #2
I am not brilliant at math, so I don't know if there is a systematic way to solve these problems. You know that [tex]t_1 = 1[/tex], so when you calculate [tex]t_2[/tex] and so on, you would plug 1 in for [tex]t_1[/tex]. Don't. Treat all cases of [tex]t_1[/tex] as if it were a symbol from another planet. Write out a few terms in terms of [tex]t_1[/tex] and see if you can find a pattern.
 
  • #3
Ok. After much confusion, I'm finally getting this straight. If you start writing out your t_n's you'll see that you are getting ratios of consecutive fibonacci numbers. This suggests that you look for solutions of the form t_n=F_(n+1)/F_n. Put this in and find a recurrence relation for the F's. (It should look familiar). Now you want to look for solutions to this recurrence relation of the form C*a^n. Using your recurrence relation, what value(s) can 'a' have? Can you add two solutions of this form? Do you have enough constants to reproduce t1=1, t2=2?
 
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  • #4
ha, yeah, you got the right idea. You need to solve a recursion.

actually, let me tell you something about a more general recursion:
[tex]x_{n+1}=\frac{a+bx_n}{c+dx_n}[/tex]

How would you solve it?
well our goal is add constants to both sides to make
[tex]x_{n+1}+\alpha=a'\frac{1+\alpha x_n}{c+dx_n}[/tex]
and
[tex]x_{n+1}+\beta=b'\frac{1+\beta x_n}{c+dx_n}[/tex]

so that when I divide the first equation by the second equation, I'll get:
[tex]\frac{x_{n+1}+\alpha}{x_{n+1}+\beta}=\frac{a'}{b'}\left (\frac{1+\alpha x_n}{1+\beta x_n}\right )[/tex]

a' and b' can be expressed in a,b,c,d, and alpha and beta respectively. Basically, all you need to do is find two solutions to a quadratic equation, if there are two distinct roots, then you are in business. The original problem can be solved using the exact method.

and of course, you can just guess and check and prove that the formula works by induction, and it will turn out to be some Fibonacci numbers...
 
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Q: What are continued fractions and why are they important?

A continued fraction is a mathematical expression that represents a real number as a sequence of fractions. They are important because they provide an alternative way to represent and approximate numbers, and are used in various fields such as number theory, cryptography, and physics.

Q: How do I find the explicit formula for a continued fraction?

To find the explicit formula for a continued fraction, you can use the algorithm known as the Gauss-Kuzmin-Wirsing constant. This algorithm involves finding the sequence of partial quotients and then using these values to construct the explicit formula.

Q: Can continued fractions be used to solve equations?

Yes, continued fractions can be used to solve certain types of equations, such as quadratic equations. They can also be used to find the roots of polynomials and to approximate solutions to transcendental equations.

Q: What is the relationship between continued fractions and infinite series?

Continued fractions and infinite series are closely related. In fact, an infinite continued fraction can be represented as an infinite series, and vice versa. Both representations involve an infinite number of terms and can be used to approximate real numbers.

Q: Are there any real-world applications of continued fractions?

Yes, there are several real-world applications of continued fractions. Some examples include using continued fractions to approximate irrational numbers in financial calculations, using them in signal processing to compress data, and using them in machine learning algorithms to improve accuracy and efficiency.

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