TITLE: Euler-Maclaurin Summation Formula for Multiple Sums
QUESTION [15 upvotes]: The Euler-Maclaurin summation formula is
\begin{eqnarray}
\sum_{k = a}^{b} f(k)  = \int_{a}^{b} f(t) \, dt  + B_1 (f(a) + f(b)) + \sum_{n = 1}^{N} \frac{B_{2n}}{(2n)!} ( f^{(2n-1)}(b) -  f^{(2n-1)}(a) )  + R_{N},
\end{eqnarray}
where $B_{n}$ is the $n^{\text{th}}$-Bernoulli number taking $B_{1} = \tfrac{1}{2}$, and the remainder term is bounded by the following
\begin{align}
|R_{N}| \leq    \frac{|B_{2N} |}{(2n)!}  \int_{a}^{b} | f^{(2N)}(t) | \, dt.
\end{align}
for any arbitrary positive integer $N$. Is there a similar formula for nested sums of the form,
\begin{eqnarray}
\sum_{k_1 = a_1}^{b_1} \cdots \sum_{k_n = a_n}^{b_n} f(k_1, \dots, k_n).
\end{eqnarray}
Thanks!

REPLY [12 votes]: Yes! There's a whole chapter about it in this book.