1.3.3. Procedures as General Methods
Exercise 1.37

Finite continued fraction

a. An infinite continued fraction is an expression of the form


f =         N₁
        -------------
        D₁ +     N₂
             -----------
             D₂ +   N₃
                -----------
                D₃ + ...

As an example, one can show that the infinite continued fraction expansion with the Ni and the Di all equal to 1 produces 1/φ, where φ is the golden ratio (described in section 1.2.2). One way to approximate an infinite continued fraction is to truncate the expansion after a given number of terms. Such a truncation -- a so-called k-term finite continued fraction -- has the form


      N₁
---------------
D₁ +   N₂
    -----------
    +     Nk
        -------
          Dk

Suppose that n and d are procedures of one argument (the term index i) that return the Ni and Di of the terms of the continued fraction. Define a procedure cont-frac such that evaluating (cont-frac n d k) computes the value of the k-term finite continued fraction. Check your procedure by approximating 1/φ using


(cont-frac (lambda (i) 1.0)
           (lambda (i) 1.0)
           k)

or successive values of k. How large must you make k in order to get an approximation that is accurate to 4 decimal places?

b. If your cont-frac procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.



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