  We use the displacement operator to derive an infinite series of integer
order derivatives for the Gr\"{u}nwald-Letnikov fractional derivative and show
its correspondence to the Riemann-Liouville and Caputo fractional derivatives.
We demonstrate that all three definitions of a fractional derivative lead to
the same infinite series of integer order derivatives. We find that functions
normally represented by Taylor series with a finite radius of convergence have
a corresponding integer derivative expansion with an infinite radius of
convergence. Specifically, we demonstrate robust convergence of the integer
derivative series for the hyperbolic secant (tangent) function, characterized
by a finite radius of convergence of the Taylor series $R=\pi/2$, which
describes bright (dark) soliton propagation in non-linear media. We also show
that for a plane wave, which has a Taylor series with an infinite radius of
convergence, as the number of terms in the integer derivative expansion
increases, the truncation error decreases. Finally, we illustrate the utility
of the truncated integer derivative series by solving two linear fractional
differential equations, where the fractional derivative is replaced by an
integer derivative series up to the second order derivative. We find that our
numerical results closely approximate the exact solutions given by the
Mittag-Leffler and Fox-Wright functions. Thus, we demonstrate that the
truncated expansion is a powerful method for solving linear fractional
differential equations, such as the fractional Schr\"{o}dinger equation.
