  In this paper, we establish a generalized Taylor expansion of a given
function $f$ in the form
  $\displaystyle{f(x) = \sum_{j=0}^m
c_j^{\alpha,\rho}\left(x^\rho-a^\rho\right)^{j\alpha} + e_m(x)}$
  \noindent with $m\in \mathbb{N}$, $c_j^{\alpha,\rho}\in \mathbb{R}$, $x>a$
and $0< \alpha \leq 1$. In case $\rho = \alpha = 1$, this expression coincides
with the classical Taylor formula. The coefficients $c_j^{\alpha,\rho}$,
$j=0,\dots,m$ as well as an estimation of $e_m(x)$ are given in terms of the
generalized Caputo-type fractional derivatives. Some applications of these
results for approximation of functions and for solving some fractional
differential equations in series form are given in illustration.
