 In this paper, the authors generalize several theorems of classical power series to fractional power series. They show that certain theorems can be derived from caputo-fractional derivatives under certain conditions. The authors then construct a new Taylor's power series with caputo-fractional derivatives. Additionally, they provide applications such as approximating fractional derivatives and integrals, solving linear and nonlinear fractional differential equations, and deriving recurrence relations for the coefficients of the fractional power series. This article was authored by Ahmad El-Aju, Omar Abu-Arqab, Ziyad Al-Zour and others.