Advantages of Continued Fraction
Continued fraction is a very useful tool. Many iterations-algorithms can be expressed easily using continued fraction. Since the convergence of continued fraction and many properties of continued fraction have been found, we can easily transform our problem in such particular algorithms into continued fraction. This transformation will allow us to derive better properties of the algorithm and use the existing mathematical methods to get better understanding of the algorithm. In other words, we do not need to reinvent the wheel because the mathematicians have already gotten the wheel for us.
Second reason to use continued fraction is the fast convergence of continued fraction, especially if the value of is large. Lange (1998) report that continued fraction can converge where power series diverge and where both type of expansion converge, continued fraction often converge faster.
Next, continued fraction is useful to approximate the real number as ratio of two integers , compute many digits of PI , euler number and other irrational numbers.
Preferable reference for this tutorial is
Teknomo, Kardi (2015) Continued Fraction. http://people.revoledu.com/kardi/tutorial/ContinuedFraction/index.html