Thermodynamical properties of spin-S Ising chains can nowadays be easily obtained using numerical calculation. However, from a mathematical point of view, its exact solution for arbitrary spin is still a challenge. Only limiting cases have been solved exactly, such as the infinite spin limit and lowest spin values. The present article addresses this issue. Using the high-temperature series expansion we obtain a new analytical series expansion of the partition function for the Ising chain, in the absence of magnetic field. Our general results cover all spins from 12 to infinite, in the high-temperature region, up to order β40(β=(kT)-1). In order to extend our results to finite-temperature we employ the method presented in the work of Ref. [Bernu and Misguich, Phys. Rev. B63 (2001) 134409]. We also present a matrix formulation of our series expansion and relate it to the transfer matrix technique.