Continued Fraction Expansion Problems
Hello All, I have some questions regarding Continued Fraction Expansion.
As an initial reference, these problems have come from my textbook in the "additional examples" area. My book doesn't really give the best explanation of those examples all the time, and as a result I have a few questions.
Q1. The numbers a_k can be found for 113/50 by using a continued fraction algorithm. Note that 113/50 is rational, and as a result it will have to terminate. Can anyone help me find this a_k ?
Q2. Given the following, one should be able to form a continued fraction expansion for .
We initially write (x,y,z,....) for the continued fraction of . From here, the continued fraction expansion for is . Knowing this, how many terms are necessary in order to accurately get 4 decimal places (2.718) ?
Q3. Assume that x=Sqrt-1. It can be proven that x = 1/(1+(1/(2+x))). Can somebody prove this? After it has been proven, the proof can be used to find the continued fraction expansion for x (Can somebody show this?). Next, find the continued franction expansion for Sqrt. Once completed, a good accuracy check is that the first 6 or 7 terms of the expansion should give a reasonable approximation for Sqrt. (The underlined portions are the questions I am asking).
Q4. What is the continued fraction expansion of Sqrt? My book notes that it might be helpful if "you find the continued fraction explansion of x where 0<x<1 and x is related to Sqrt.
Q5. What is the continued fraction expansion of Sqrt? My book notes that part of the expansion is (2,1,1,1,4,1,1,1,...). Is the next entry in this expansion a 2 or a 4 ? Also, it mentions that you can find some number like what I had mentioned in Q4, and once we have our answer we should be able to show why it is correct.
Any help is appreciated!