What is the proper notation for a general number of nested summations?

Question

A sum over one index: $\sum_i f(i)$

A sum over two indices: $\sum_i \sum_j f(i,j)$

A sum over many indices: $\sum_{k_1} \sum_{k_2} \underbrace{\dots}_n \sum_{k_n} f(\mathbf k)$?

Answer 1

Try \begin{equation} \sum_{x \mathop \in S} f(x) \end{equation} which is the sum of $f(x)$ over all elements $x$ in the set $S$. Your set $S$ could be $\{k_{1},...,k_{n}\}$

Answer 2

Probably in most contexts $$\sum_{k_1} \cdots \sum_{k_n} f(k_1, \ldots, k_n)$$ suffices, and sometimes one will just write $$\sum_{k_1, \ldots, k_n} f(k_1, \ldots, k_n),$$ or perhaps slightly more precisely, $$\sum_{(k_1, \ldots, k_n)} f(k_1, \ldots, k_n).$$ As written, this last summation isn't actually a multiple summation, but rather a single summation over all $k$-tuples $(k_1, \ldots, k_n)$ in the Cartesian product $K_1 \times \cdots \times K_n$ of the index sets $K_i$ over which the index variables $k_i$ respectively vary.

As usual, one must take care when dealing with infinite sums, which formally are different objects, and for which it is possible that the summation depends on the order of indexing. In this setting, the first notation above is surely best; if you must use one of the latter notations, a comment remarking on the order of summation (or an indication that the sum is independent of the indexing order) would often be prudent.