Eikonal Blog

2010.02.12

Some continued fractions

Filed under: mathematics — Tags: — sandokan65 @ 21:01
  • \sqrt{2} = 1+ \frac1{2+}\frac1{2+}\frac1{2+}\cdots = [1;2,2,2,\cdots] = [1;\dot{2}].
  • \sqrt{3} = [1;\dot{1},\dot{2}].
  • \sqrt{5} = [2; \dot{4}].
  • \sqrt{7} = [2; \dot{1},1,1,\dot{4}].
  • e = [2;1,2,1,1,4,1,1,6,1,1,8,\cdots].
  • \frac\pi4 = \frac1{1+} \frac{1^2}{2+} \frac{3^2}{2+} \frac{5^2}{2+} \frac{7^2}{2+} \frac{9^2}{2+} \cdots.
  • [3;\dot{3}] = \frac{3+\sqrt{13}}2 = 3.302,77\cdots.
  • [a;\dot{b}] = a+\frac{\sqrt{b^2+4}-b}2.
  • [4;\dot{6}] = 1+\sqrt{10}.
  • [a;\dot{b},\dot{c}] = ab + \frac{\sqrt{bc(bc+4)}-bc}2.

Theorem: A simple infinite continued fraction [a_0; a_1, a_2, \cdots] converges IFF \sum_n a_n diverges.

Theorem:
Every real number x has the unique simple continued fraction expression. That expression is:

  • finite IFF x \in {\Bbb Q},
  • infinite and repeating IFF x \not\in {\Bbb Q} but algebraic,
  • infinite and non-repeating IFF x is transcendental.

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