TITLE: An infinite series $1+\frac{1}{2+\frac{1}{3+\frac{1}{4+.......}}}$
QUESTION [5 upvotes]: How can we find the value to which the following series converges, if it converges to a finite number? If else, how can we prove that it is divergent?$$1+\frac{1}{2+\frac{1}{3+......}}$$

REPLY [2 votes]: We can see that
$$
1 < 1+\frac{1}{2+\frac{1}{3+......}}
$$
and
$$
\frac{1}{2} > \frac{1}{2+\frac{1}{3+......}}
$$
so
$$
1 < 1+\frac{1}{2+\frac{1}{3+......}} < \frac{3}{2}
$$