Chain sequence - Alchetron, The Free Social Encyclopedia

In the analytic theory of continued fractions, a chain sequence is an infinite sequence {a_n} of non-negative real numbers chained together with another sequence {g_n} of non-negative real numbers by the equations

a 1 = ( 1 − g 0 ) g 1 a 2 = ( 1 − g 1 ) g 2 a n = ( 1 − g n − 1 ) g n

where either (a) 0 ≤ g_n < 1, or (b) 0 < g_n ≤ 1. Chain sequences arise in the study of the convergence problem – both in connection with the parabola theorem, and also as part of the theory of positive definite continued fractions.

The infinite continued fraction of Worpitzky's theorem contains a chain sequence. A closely related theorem shows that

f ( z ) = a 1 z 1 + a 2 z 1 + a 3 z 1 + a 4 z ⋱

converges uniformly on the closed unit disk |z| ≤ 1 if the coefficients {a_n} are a chain sequence.

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 {g_n} = {x}, and x < ½, the resulting sequence {a_n} will be an endless repetition of a real number y that is less than ¼.

The choice g_n = ½ is not the only set of generators for this particular chain sequence. Notice that setting

generates the same unending sequence {¼, ¼, ¼, ...}.

References

Chain sequence Wikipedia

(Text) CC BY-SA