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.


sqrt2_a(n) ; function definition is as simple as that
{
return n?2.0:1.0
}
 
sqrt2_b(n)
{
return 1.0
}
 
napier_a(n)
{
return n?n:2.0
}
 
napier_b(n)
{
return n>1.0?n-1.0:1.0
}
 
pi_a(n)
{
return n?6.0:3.0
}
 
pi_b(n)
{
return (2.0*n-1.0)**2.0 ; exponentiation operator
}
 
calc(f,expansions)
{
r:=0,i:=expansions
; A nasty trick: the names of the two coefficient functions are generated dynamically
; a dot surrounded by spaces means string concatenation
f_a:=f . "_a",f_b:=f . "_b"
 
while i>0 {
; You can see two dynamic function calls here
r:=%f_b%(i)/(%f_a%(i)+r)
i--
}
 
return %f_a%(0)+r
}
 
Msgbox, % "sqrt 2 = " . calc("sqrt2", 1000) . "`ne = " . calc("napier", 1000) . "`npi = " . calc("pi", 1000)