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Other name: Cyclic continued fraction; recurring chain fraction

Periodic continued fraction is defined as a infinite continued fraction where for some fixed integer and for all . We put bar notation to represent the part of the sequence that repeated infinitely.

Notation is the periodic part that repeated infinitely, thus

The cyclic continued fraction is proven convergence (see Phillips, 2000 )

Step by step to compute the convergence limit of periodic continued fraction analytically is as follow

1. Identify the periodic terms and set it as

2. Compute the periodic term as continued fraction using difference equation , and get the ratio of the last two convergent

and

3. Solve quadratic equation and select positive

4. Compute continued fraction using difference equation above.

## Example

Calculate cyclic continued fraction !

The periodic terms are . To compute the convergence of this periodic term we set , , , . Then we set quadratic equation , or , or . Factoring the quadratic equation, we have positive . Then we compute continued fraction = . Using the same difference equation again we have , , , . Thus, the convergence limit of the continued fraction is

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Preferable reference for this tutorial is

Teknomo, Kardi (2015) Continued Fraction. http://people.revoledu.com/kardi/tutorial/ContinuedFraction/index.html