Use a continued fraction to estimate $e$

Question

I am working on the following question "Use the continued fraction $[1;0,1,1,2,1,1,4,1,1,6,1,1]$ to get an estimate for $e$." But I got stuck when I tried to compute $q_i$, since $a_1=0$ , $q_1 =0$. Therefore, $S_1$ is undefined

Answer 1

It is quite "natural" that early convergences are "problematic". In fact, $S_{-1}$ is always $\frac{1}{0}$. (Sometimes people define $p_{-2}=0$ and $q_{-2}=1$, so $S_{-2} \equiv 0$) $q_1=0$ is not a big problem, since you notice that $q_2 = a_2q_1+q_0 = 1 \cdot 0 + 1 = 1$ and all following $q_k$ are good.

Fact: $S_k$ is a better approximation than $S_n ~\forall ~n<k$. So it is expected that sometimes $S_1$ is way too far from a good approximation.