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