The general expression of the Fluctuationlessness Theorem states that the matrix representation of an algebraic operator which multiplies its argument by a scalar univariate function, is identical to the image of the independant variable's matrix representation over the same subspace via the same basis set, under that univariate function, when the fluctuation terms are ignored. Just by using this basic idea, this principle applied on the remainder term of a Taylor expansion a highly versatile approximation can be obtained. Taking into consideration that Euler method and higher order Taylor methods are using Taylor expansion to solve IVP problems, we applied the Fluctuationlessness theorem to the remainder term of Taylor expansion expressed in integral form and adapted it to Euler method by extending it via added remainder term and obtain an additional effectiveness in approximation.