The integral test of convergence states that, if $f:[1,+\infty)\to[0,+\infty)$ is a monotonically decreasing nonnegative function, then the series $\sum_1^\infty f(n)$ converges iff $\int_1^\infty f(n) dn$ is finite.
Is the high-dimensional generalization also true? That is, given $f:[1,+\infty)^N \to[0,+\infty)$, and $f(\dotsc,n_i,\dotsc) \ge f(\dotsc,n_i+\epsilon,\dotsc)$ for all $1\le i\le N$ and $n_i\in[1,+\infty)$ and $\epsilon>0$, then the sum $$ \sum_{n_1=1}^\infty \cdots \sum_{n_N=1}^\infty f(n_1,\dotsc,n_N) $$ converges iff the multiple integral $$ \int_1^\infty \cdots \int_1^\infty f(n_1,\dotsc,n_N) dn_1 \dotsm dn_N $$ is finite.
(This is just for checking if my answer over physics.SE is reasonable.)