Chain Sequence (continued Fraction) - An Example

An Example

The sequence {¼, ¼, ¼, ...} appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting g0 = g1 = g2 = ... = ½, it is clearly a chain sequence. This sequence has two important properties.

  • Since f(x) = x − x2 is a maximum when x = ½, this example is the "biggest" chain sequence that can be generated with a single generating element; or, more precisely, if {gn} = {x}, and x < ½, the resulting sequence {an} will be an endless repetition of a real number y that is less than ¼.
  • The choice gn = ½ is not the only set of generators for this particular chain sequence. Notice that setting
g_0 = 0 \quad g_1 = {\textstyle\frac{1}{4}} \quad g_2 = {\textstyle\frac{1}{3}} \quad g_3 = {\textstyle\frac{3}{8}} \;\dots
generates the same unending sequence {¼, ¼, ¼, ...}.

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