The influence of elasticity on pulsed Taylor-Couette flow in a linear Maxwell fluid is investigated. We consider the case, in which the inner cylinder is oscillating with a periodic angular velocity, \ \backslashOmega_\0\\backslashcos(\backslashomega t)\ , and the outer cylinder is fixed. Attention is focused on the linear stability analysis which is solved using the Floquet theory and a technique of converting a boundary value problem to an initial value problem. Results obtained in this framework show that, in the high-frequency limit, the Deborah number has a destabilizing effect and the critical Taylor and wave numbers tend toward constant values independently of the frequency number. However, in the low-frequency limit, the Maxwell fluid behaves as a Newtonien one and the Deborah number has no effect on the stability of the basic state which tends to the classical configuration of steady circular Couette flow. These numerical results are in good agreement with the asymptotic analysis performed in the limit of low and high frequencies.