Non convergent simple continued fractions?

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.

(This is theorem 10 (p.10–12) in Continued Fractions, A. Ya. Khinchin, Phoenix Science Press 1964, or theorem 9.31 (p.185) in Neverending Fractions: An Introduction to Continued Fractions by Borwein et al., Cambridge University Press 2014.)