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

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 $$a_i=1$$

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

• $$b_i=0$$ "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.