A general form of the generalized Taylor's formula with some applications

•In this article, a new general form of fractional power series is introduced.•Some results of classical power series are circulated and proved to this new form.•A new general form of the generalized Taylor's formula is also obtained.•Applications including fractional power series solutions for FDEs are analyzed.•Results reveal that, the new expansion is effective for formulating the solutions.

In this article, a new general form of fractional power series is introduced in the sense of the Caputo fractional derivative. Using this approach some results of the classical power series are circulated and proved to this fractional power series, whilst a new general form of the generalized Taylor's formula is also obtained. Some applications including fractional power series solutions for higher-order linear fractional differential equations subject to given nonhomogeneous initial conditions are provided and analyzed to guarantee and to confirm the performance of the proposed results. The results reveal that the new fractional expansion is very effective, straightforward, and powerful for formulating the exact solutions in the form of a rapidly convergent series with easily computable components.