  The Euler--Maclaurin (EM) summation formula is used in many theoretical
studies and numerical calculations. It approximates the sum $\sum_{k=0}^{n-1}
f(k)$ of values of a function $f$ by a linear combination of a corresponding
integral of $f$ and values of its higher-order derivatives $f^{(j)}$. An
alternative (Alt) summation formula was recently presented by the author, which
approximates the sum by a linear combination of integrals only, without using
high-order derivatives of $f$. It was shown that the Alt formula will in most
cases outperform, or greatly outperform, the EM formula in terms of the
execution time and memory use. In the present paper, a
multiple-sum/multi-index-sum extension of the Alt formula is given, with
applications to summing possibly divergent multi-index series and to sums over
the integral points of integral lattice polytopes.
