Necessary and sufficient conditions for convergence of integer continued fractions

Fundamental to the theory of continued fractions is fact that every infinite fraction with positive integer coefficients converges; however, it unknown precisely which (not necessarily positive) converge. Here we present a simple test determines whether an converges or diverges. In addition, for convergent specifies limit rational irrational. An attractive way visualise model them as paths on Farey graph, graph embedded in hyperbolic plane induces tessellation by ideal triangles. With this geometric representation our convergence can be interpreted particularly elegant manner, giving deeper insight into nature convergence.