A function's Taylor series is an infinite term representation based on denumerable infinite number of information given at a point we call expansion point. These information are the values of the function's derivatives at the expansion point. On the other hand, Taylor expansion formula for a function contains finite number term containing finite number of information about the function plus a remainder term which somehow gathers all remaining denumerable information into an integral over the function. The integral, in its other name, the remainder is generally ignored and the remaining polynomial part is used as an approximation. Our aim in this paper is to represent the remainder term of Taylor expansion by means of the recently developed fluctuation free integration. Gaussian wave type basis functions are used as the elements of the spanning set. Calculations are done using several values for the parameters of the basis. They show the performance characteristics of the presented method and the results are sometimes overwhelming the other methods and sometimes promising.