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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
dare 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-fracsuch 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
kin order to get an approximation that is accurate to 4 decimal places?
cont-fracprocedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.