Abstract
Abstract In this paper, We give a generalization the resut of Roger B. Nelsen, by giving a closed form expression for x = [ a 0 , a 1 , ⋯ , a k , b 1 , ⋯ b m ¯ ] , $\left[ {{a}_{0}},{{a}_{1}},\cdots ,{{a}_{k}},\overline{{{b}_{1}},\cdots {{b}_{m}}} \right],$
Highlights
The algorithm stops after finitely many steps if and only if x is rational
If there exists k ≥ 0 and m > 0 such that whenever r > k, we have ar = ar+m, the continued fraction is said periodic, with period (b1, · · ·, bm) = and pre-period (a0, a1, · · ·, ak), which can be written for simplicity x = [a0, a1, · · ·, ak, b1, · · ·, bm]
We find a closed form expression for x = [a0, a1, · · ·, ak, b1, · · ·, bm], which generalized a previous resut of Roger B
Summary
The algorithm stops after finitely many steps if and only if x is rational. The above expansion is called The simple continued fraction of x. Abstract In this paper, We give a generalization the resut of Roger B. By giving a closed form expression for x = [a0, a1, · · · , ak, b1, · · · , bm], Keywords: Continued fractions, periodic, proof without words.
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