The main goal of this study is to find the solutions of linear fractional differential equations of order 2q, including small delay, where 0 < q < 1 which has various applications. The fractional derivatives are taken in the sense of Caputo which is more suitable than Riemann-Liouville sense. We assume that the order q satisfy the condition nq = 1 for some natural number n which determines the number of the linearly independent solutions. Since the delay term is small, the linear fractional differential equation is expanded in powers series of which reduce the problem to regular or singular perturbation problem for which it is easier to find the solution. The solution is obtained in the form of a series expansion of E. To demonstrate the accuracy and the effectiveness of the proposed approach, some illustrative examples are presented.