# Non convergent simple continued fractions?

Let $$$$ be an infinite sequence of integers such that $$00.$$

For any natural $$n$$ we know there exists a convergent: one rational number $$r_n$$ such that it is equal to the simple continued fraction $$r_n=a_0+\frac{1}{a_1 +\frac{1}{a_2 +\frac{1}{\ddots_{a_n+0}}}}$$ $$r_n$$ is the number given by the finite continued fraction of the original sequence truncated at $$n$$, $$$$.

The question is: given any sequence of positive integers, does the simple continued fraction always converge to a real number?

In fact, an even better theorem is available, the Seidel-Stern theorem. Let $$\langle a_i\rangle$$ be any sequence of positive real numbers. Then the continued fraction $$[a_0; a_1, a_2, \ldots]$$ converges if and only if the series $$\sum_{i=0}^\infty a_i$$ diverges!
By restricting the $$a_i$$ to be positive integers, we get the answer to your question: all such continued fractions converge.