Generalized Taylor’s formula

In this paper, a new generalized Taylor’s formula of the kindf(x)=∑j=0naj(α)(x-a)jα+Rnα(x),where aj ∈ R, x > a, 0 < α ⩽ 1, is established. Such an expression is precisely the classical Taylor’s formula in case of α = 1. In addition, detailed expressions for Rnα(x) and aj(α), that involving Caputo fractional derivatives, are well constructed. Some applications including approximation of functions and solutions of fractional differential equations are also given.