The analytic theory of continued fractions is used to discuss the dynamic properties of a system described by a tridiagonal eigenvalue equation, equivalent to a one-dimensional tight-binding Hamiltonian in which only the nearest-neighbour hopping energy coefficients are finite. Expressions for the density of states and the dynamic susceptibility are reported. The expressions for these response functions reduce to simple closed forms when the coefficients in the eigenvalue equation are periodic functions. A model of linear transverse spin fluctuations in a longitudinally modulated magnet is chosen to motivate the analytic work, and an example of the spectrum of spontaneous spin fluctuations is provided together with the corresponding Lyapunov exponents.