Convergence of a finite continued fraction for $b_i \in [-1,0]$ and $a_i=1$

Question

based on wiki page here, finite continued fraction is as follows:

$$a_0+\cfrac{b_0}{a_1+\cfrac{b_1}{\ddots+\cfrac{\ddots}{a_{n-1}+\cfrac{b_{n-1}}{...}}}}$$

I want to find the limit of finite continued fraction for $b_i\in [-1,0]$ and $a_i=1$

$$1+\cfrac{b_0}{1+\cfrac{b_1}{\ddots+\cfrac{\ddots}{1+\cfrac{b_{n-1}}{...}}}}$$

Answer 1

To begin with, I think it should be noted that

• $b_i=0$ "stops" the continued fraction in its track, so it would probably be best to ignore $0$, except perhaps for $b_0$.
• also, the final term $b_n$ better not be $-1$ (I don't know about you, but I really don't like going anywhere near the divide-by-zero monster)

With that out of the way, I can't say that I clearly understand the notion of limit in this context. A finite continued fraction simply results in a number. If we're interested in upper/lower bounds on what numbers can be represented with this set of rules, a couple examples seem to suggest that there aren't any. For instance with $$ [-1,b_1] = 1-\cfrac{1}{1+b_1} = \cfrac{b_1}{1+b_1} $$ it is clear that the closer to $-1^+$ we choose $b_1$, the more negative the value will become, tending to $-\infty$. Similarly, with $$ [-1,-1,b_2] = 1-\cfrac{1}{1-\cfrac{1}{1+b_2}} = -\cfrac{1}{b_2} $$ it is also clear that the closer $b_2$ approaches $0^-$, the larger the value gets, tending to $+\infty$ this time.