Exercise 1.37

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

    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

    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)

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

  2. 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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