We will begin with a general survey of decision problems related to the orbits and invariants of linear dynamical systems and how these problems relate to properties of recurrence sequences. This section will focus on principal techniques, open problems, and current challenges in the field. The second part of these lectures will narrow our focus to second-order P-finite sequences. Recall that a sequence is P-finite if it satisfies a linear recurrence relation with polynomial coefficients. We will connect the decision problems for second-order P-finite sequences to the convergence of polynomial continued fractions.