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Natural Logarithm Approximated by Continued Fractions

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number of terms
1
x range
y range
​
natural log
power (Taylor) series
CF before contraction
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.
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