Continued fraction

A number may be represented as a continued fraction (see Mathworld for more information) as follows:

Continued fraction is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

$cf=a_{1}+{\cfrac {b_{1}}{a_{2}+{\cfrac {b_{2}}{a_{3}+{\cfrac {b_{3}}{\mathrm {etc\ldots } }}}}}}$

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.

For the square root of 2, $b_{N}$ is always $1$. $a_{1}$ is $1$ then $a_{N}$ is $2$.

For Napier's Constant, $a_{1}$ is $2$, then $a_{N}$ is $N-1$. $b_{1}$ is $1$ then $b_{N}$ is $N-1$.

For Pi, $a_{1}$ is 3 then $a_{N}$ is always $6$. $b_{N}$ is $(2N-1)^{2}$.

Python

<lang python># The continued Fraction def CF(a, b, t):

 if 0 < t:
   a1 = next(a)
   b1 = next(b)
   z  = CF(a, b, t-1)
   return (a1*z[0] + b1*z[1], z[0])
 return (1,1)

1. Convert the continued fraction to a string

def pRes(cf, d):

 res = str(cf[0] // cf[1])
 res += "."
 x = cf[0] % cf[1]
 while 0 < d:
   x *= 10
   res += str(x // cf[1])
   x = x % cf[1]
   d -= 1
 return res

1. Test the Continued Fraction for sqrt2

def sqrt2_a():

 yield 1
 while (True):
   yield 2

def sqrt2_b():

 while (True):
   yield 1

cf = CF(sqrt2_a(), sqrt2_b(), 950) print(pRes(cf, 200))

1. 1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147

1. Test the Continued Fraction for Napier's Constant

def Napier_a():

 yield 2
 n = 1
 while (True):
   yield n
   n += 1

def Napier_b():

 n=1
 yield n
 while (True):
   yield n
   n += 1

cf = CF(Napier_a(), Napier_b(), 950) print(pRes(cf, 200))

1. 2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852516642742746639193200305992181741359662904357290033429526059563073813232862794349076323382988075319525101901

1. Test the Continued Fraction for Pi

def Pi_a():

 yield 3
 while (True):
   yield 6

def Pi_b():

 x = 1
 while (True):
   yield x*x
   x += 2

cf = CF(Pi_a(), Pi_b(), 950) print(pRes(cf, 10))

1. 3.1415926532</lang>