Euler expressed certain sums of the form
\sum_{k=1}^\infty \Bigl(1 + {1 \over 2^m} + \cdots +
{1 \over k^m}\Bigr) (k + 1)^{-n}\hbox{,}
¶ where m and n are positive integers, in terms of the Riemann zeta
function. In [Borwein et al.\ 1993], Euler's results were extended to
a significantly larger class of sums of this type, including sums with
alternating signs.
¶ This research was facilitated by numerical computations using an
algorithm that can determine, with high confidence, whether or not a
particular numerical value can be expressed as a rational linear
combination of several given constants. The present paper presents
the numerical techniques used in these computations and lists many of
the experimental results that have been obtained.