  Define the monomials $e_n(x) := x^n$ and let $L$ be a linear functional. In
this paper we describe a method which, under specified conditions, produces
approximations for the value $L(e_0 )$ in terms of Hankel determinants
constructed from the values $L(e_1 )$, $L(e_2 )$, . . . . Many constants of
mathematical interest can be expressed as the values of integrals. Examples
include the Euler-Mascheroni constant $\gamma$, the Euler-Gompertz constant
$\delta$, and the Riemann-zeta constants $\zeta(k)$ for $k \ge 2$. In many
cases we can use the integral representation for the constant to construct a
linear functional for which $L(e_0)$ equals the given constant and $L(e_1)$,
$L(e_2)$, . . . are rational numbers. In this case, under the specified
conditions, we obtain rational approximations for our constant. In particular,
we execute this procedure for the previously mentioned constants $\gamma$,
$\delta$, and $\zeta(k)$. We note that our approximations are not strong enough
to study the arithmetic properties of these constants.
