In this paper, first of all, we defined the generalized order and generalized type of Taylor entire function; Secondly, we show some interesting relationship on the maximum modulus, the maximum term and the coefficients of Taylor entire function; Finally, we study the polynomial approximation of entire functions in Banach spaces ((B(p, q, k); f ), Hardy spaces, Bergman spaces), the coefficient characterization of generalized type of Taylor entire function of slow growth has been obtained in terms of the approximation errors.