An infinite series $1+\frac{1}{2+\frac{1}{3+\frac{1}{4+.......}}}$

Question

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?$$1+\frac{1}{2+\frac{1}{3+......}}$$

Answer 1

We can see that $$ 1 < 1+\frac{1}{2+\frac{1}{3+......}} $$ and $$ \frac{1}{2} > \frac{1}{2+\frac{1}{3+......}} $$ so $$ 1 < 1+\frac{1}{2+\frac{1}{3+......}} < \frac{3}{2} $$