  In this paper we construct approximations for the Caputo derivative of order
$1-\alpha,2-\alpha,2$ and $3-\alpha$. The approximations have weights
$0.5\left((k+1)^{-\alpha}-(k-1)^{-\alpha}\right)/\Gamma(1-\alpha)$ and
$k^{-1-\alpha}/\Gamma(-\alpha)$, and the higher accuracy is achieved by
modifying the initial and last weights using the expansion formulas for the
left and right endpoints. The approximations are applied for computing the
numerical solution of ordinary fractional differential equations. The
properties of the weights of the approximations of order $2-\alpha$ are similar
to the properties of the $L1$ approximation. In all experiments presented in
the paper the accuracy of the numerical solutions using the approximation of
order $2-\alpha$ which has weights $k^{-1-\alpha}/\Gamma(-\alpha)$ is higher
than the accuracy of the numerical solutions using the $L1$ approximation for
the Caputo derivative.
