Natural Logarithm Approximated by Continued Fractions

number of terms

1

x range

y range


	natural log
	power (Taylor) series
	CF before contraction
	CF after contraction

Continued fractions provide a very effective toolset for approximating functions. Usually the continued fraction expansion of a function approximates the function better than its Taylor or Fourier series. This Demonstration compares the quality of three approximations to

ln(1+x)

. One is the Taylor series and the other two are continued fraction expansions. The first continued fraction expansion can be obtained as a canonical even contraction of a continued fraction using Euler's method to transform a series to an

S

-fraction. The other continued fraction expansion was developed by the author as a canonical even contraction from the first one.