  We investigate the exact evolution of the reduced dynamics of a one qubit
system as central spin coupled to a femionic layered environment with unlimited
number of layers. Also, we study the decoherence induced on central spin by
analysis solution is obtained in the limit $N\rightarrow\infty$ of an infinite
number of bath spins. Finally, the Nakajima-Zwanzig (NZ) and the
time-convolutionless (TCL) projection operator techniques to second order are
derived.
