continued fraction
Mathworld





a

0


+









b

1












a

1


+









b

2












a

2


+









b

3












a

3


+
⋱

















{\displaystyle a_{0}+{\cfrac {b_{1}}{a_{1}+{\cfrac {b_{2}}{a_{2}+{\cfrac {b_{3}}{a_{3}+\ddots }}}}}}}


The task is to write a program which generates such a number and prints a real representation of it. The code should be tested by calculating and printing the square root of 2, Napier's Constant, and Pi, using the following coefficients:

For the square root of 2, use 




a

0


=
1


{\displaystyle a_{0}=1}

 then 




a

N


=
2


{\displaystyle a_{N}=2}

. 




b

N




{\displaystyle b_{N}}

 is always 



1


{\displaystyle 1}

.







2


=
1
+








1









2
+








1









2
+








1









2
+
⋱

















{\displaystyle {\sqrt {2}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots }}}}}}}


For Napier's Constant, use 




a

0


=
2


{\displaystyle a_{0}=2}

, then 




a

N


=
N


{\displaystyle a_{N}=N}

. 




b

1


=
1


{\displaystyle b_{1}=1}

 then 




b

N


=
N
−
1


{\displaystyle b_{N}=N-1}

.





e
=
2
+








1









1
+








1









2
+








2









3
+








3









4
+
⋱






















{\displaystyle e=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {2}{3+{\cfrac {3}{4+\ddots }}}}}}}}}


For Pi, use 




a

0


=
3


{\displaystyle a_{0}=3}

 then 




a

N


=
6


{\displaystyle a_{N}=6}

. 




b

N


=
(
2
N
−
1

)

2




{\displaystyle b_{N}=(2N-1)^{2}}

.





π
=
3
+








1









6
+








9









6
+








25









6
+
⋱

















{\displaystyle \pi =3+{\cfrac {1}{6+{\cfrac {9}{6+{\cfrac {25}{6+\ddots }}}}}}}




See also

   Continued fraction/Arithmetic for tasks that do arithmetic over continued fractions.


 
// I provide four functions:-
// cf2S general purpose continued fraction to sequence of float approximations
// cN2S Normal continued fractions (a-series always 1)
// cfSqRt uses cf2S to calculate sqrt of float
// π takes a sequence of b values returning the next until the list is exhausted after which  it injects infinity
// Nigel Galloway: December 19th., 2018
let cf2S α β=let n0,g1,n1,g2=β(),α(),β(),β()
             seq{let (Π:decimal)=g1/n1 in yield n0+Π; yield! Seq.unfold(fun(n,g,Π)->let a,b=α(),β() in let Π=Π*g/n in Some(n0+Π,(b+a/n,b+a/g,Π)))(g2+α()/n1,g2,Π)}
let cN2S = cf2S (fun()->1M)
let cfSqRt n=(cf2S (fun()->n-1M) (let mutable n=false in fun()->if n then 2M else (n<-true; 1M)))
let π n=let mutable π=n in (fun ()->match π with h::t->π<-t; h |_->9999999999999999999999999999M)
 