On Infinite Process Convergence, Part III: Introduction to Continued Fractions
|Title||On Infinite Process Convergence, Part III: Introduction to Continued Fractions|
|Author(s)||Thomas E Phipps|
|Keywords||Infinite Process Convergence, Continued Fractions|
The ?difference equation viewpoint? previously applied to speeding the convergence of infinite series convergent in the Cauchy sense [Phys. Essays 6, 135 (1993)] and to forcing the convergence of series divergent in that sense [Phys. Essays 6, 440 (1993)] is here applied to continued fractions. The difference equation equivalent to a continued fraction is of second order, whereas that equivalent to an infinite series is of first order. The order of its equivalent difference equation is equal to the root multiplicity of any discrete infinite process, an nth-order process being capable of assuming (in the absence of root confluence) up to n distinct ?values.? This means that continued fractions are rightly conceived at the definitional level as bivalued. We give examples to support this claim.