WOLFRAM|DEMONSTRATIONS PROJECT

Approximating Pi with Continued Fractions

​
parameter k
4
number of terms n
7
display
contraction
no contraction
absolute error
base-10 log of relative error
with no contraction
2.030603994×
-11
10
-11.18952464
with contraction
1.622146105×
-15
10
-15.28705990
convergent for π (no contraction):
65536
122517+
1
34+
9
34+
25
34+
49
34+
81
34+
121
34
=
173070024704
55089899865
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 two approximations for
π
. One is a continued fraction approximation derived from one for the Gamma function and based on that, the other is a continued fraction expansion the author has developed as a canonical even contraction.