  This paper focuses on the equivalent expression of fractional
integrals/derivatives with an infinite series. A universal framework for
fractional Taylor series is developed by expanding an analytic function at the
initial instant or the current time. The framework takes into account of the
Riemann-Liouville definition, the Caputo definition, the constant order and the
variable order. On this basis, some properties of fractional calculus are
confirmed conveniently. An intuitive numerical approximation scheme via
truncation is proposed subsequently. Finally, several illustrative examples are
presented to validate the effectiveness and practicability of the obtained
results.
