A Python file to play with a special continued fraction. If you don't have a Python interpreter, try running it in an online one (copy and paste the code), like: http://repl.it/ http://codepad.org/ http://labs.codecademy.com/

x.py

def x(k):
    """Numerically investigate the continued fraction x = 
    1 +         1
         ________________
          1 +      1
               __________
                1 +   1
                    _____
                      ...
    We won't be able to go out to infinity, so we'll use a counter to take only
    a (large) finite number of steps down this fraction.

    This is a recursive function, in that it will call itself until a certain
    stopping criterion is met.  If our counter k is less than 1, we'll return
    1; this is the stopping criterion---or base case---of our recursion.  If
    not, we'll return 1 + 1 / x(k - 1) (our recursive step).
    """
    if k <= 1:
        return 1.0
    else:
        return 1.0 + 1.0 / x(k - 1)


# Now we'll see this in action
if __name__ == '__main__':
    # Print out 250 steps of our continued fraction
    print(x(250))  # Does the output look familiar?