This letter presents a new approximation algorithm suitable for Taylor series expansion. The algorithm can effectively reduce the approximation error by using only a small number of terms in series expansion. The proposed algorithm can be more easily implemented and has better performance than the conventional Euler approximation algorithm. Also, the approximation performance of the proposed algorithm is compared with direct approaches, such as least-squares and Chebyshev approximation algorithm to show the relative advantages of the proposed algorithm. This letter also presents an application example of the proposed approximation algorithm in carrier recovery of OFDM (Orthogonal Frequency Division Multiplexing) modem.