We analyze an active mode-locked laser under the influence of asynchronous spontaneous emission (ASE) noise by using eight mathematical series Sn, Qn, Rn, an, bn, cn, rn, and Pn to trace the evolution of the noise. The series are easily calculated from the laser parameters and are used to determine the steady-state pulse of the active mode-locked laser operating in both exactly tuned and detuned conditions. Series analysis results of ideal noiseless laser models are consistent with that of classical self-consistence methods. The advantage of our series approach is that it can be used for studying laser model even in the presence of the inline optically amplified noise. Analysis of the laser model with ASE noise reveals that the large noise figure amplifier and high cavity loss degrade the signal-to-noise ratio (SNR). Large decrease of SNR caused by detuning limits the locking range of the laser. The series method can be used not only to determine the characteristics of the steady state pulse but also to study the transient process of the noise inside the laser and to determine the locking range of the laser when the ASE noise is considered. Our analytical results are visualized by simulation results.