The best way to start a continued fraction is with a fraction
A representation for the number 1 could be:
Replacing the ‘1’ in the denominator as :
Then the same again for the bottom 1:
And hence, ‘1’ can be expressed as:
(continued till infinity)
However,
A representation for the number 2 in a similar way could be:
Replacing the ‘2’ in the denominator as :
And hence, in the exact similar way, ‘2’ can be expressed as:
(continued till infinity)
Upon closer inspection we see that both of these are equal. So does that mean:
Even if we were given this continued fraction as a question:
We would solve it as:
So the question becomes, how do you find out the answer, is it 2 or is it 1 or are they both equal?
Comments
As an infinite expression, what you’re really calculating is the limit of the following sequence:
Where in one case , and in the other .
Note that this sequence can be expressed recursively by the sequence defined as therefore, the formal equivalent of “evaluate this infinite continued fraction” would be to find the limit of this sequence as varies, which you remarked is when and when .
The “true limit”, so to speak, is the one most points converge to.
Consider the function (so );
what you remarked can be formalized by saying that, if converges to a real value , since is continuous then i.e. .
Now, a result about this kind of sequence tells us that, if , then a neighborhood of has all points converging to ;
if instead , then no neighborhood of has all its points converging to .
Since and , we could say that the “true limit” would be 1.