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(Solved): (2) A finite continued fraction is: a0+a1+a2++an1111 and may be denoted ...



(2) A finite continued fraction is:
\[
a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{\ddots+\frac{1}{a_{n}}}}}
\]
and may be d

(2) A finite continued fraction is: and may be denoted by . (a) Write a program that takes as inputs and gives the floating point value of the finite continued fraction using the above definition. (b) An infinite continued fraction for the golden ratio is: A convergent is obtained by terminating a continued fraction at a finite number of operations. The convergents for the continued fraction above are: Write a program that computes successive convergents until is found accurate to five significant digits. Include a copy of your code and a table with successive convergents. (c) A continued fraction for is: Assuming that you can work with infinite continued fractions the same way that you work with finite fractions, can you give a quick argument why this is true? Modify your program for the golden ratio above to compute to five significant digits. (d) For a given finite continued fraction , define and by: and . Then the continued fraction equals . Write a program that uses this recurrence relation to compute the continued fraction expansion for any , and compare the values you get for and with those you arrived at in parts (b), (c) above.


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(a) Here is an example program in Python that takes a list of integers as input and returns the floating point value of the finite continued fraction:
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