Is there an accepted notation for the $n$th sum/integral of a function? I'm working on a thesis in image processing at the moment, and wanted to include a portion concerning the methodology for n dimension images (it will include a fully write up and functioning code for 2D and 3D images), and that would involve taking the $n$th integral of a function of $n$ variables, which in this case would mean the nth sum. For the sum portion, it would something like the sum of $x_1$ from $0$ to $M$, then the sum of $x_1$ from $0$ to $M$, and so on with a total of of $N$ summations.
Is there any sort of widely accepted convention for annotating this, or should I just put something like sum1 of sum2 /dots then the last sum?
edit: specific examples (sorry, didn't know you could use LaTeX here):
2D: $$\sum_{x_1=0}^M \sum_{x_2=0}^M f(x_1,x_2)$$
3D: $$\sum_{x_1=0}^M \sum_{x_2=0}^M \sum_{x_3=0}^M f(x_1,x_2,x_3)$$
Is there a nicer way to express something like:
$$\sum_{x_1=0}^M \sum_{x_2=0}^M \dots \sum_{x_n=0}^M f(x_1,x_2,\dots,x_n)$$

I don't know of a convention. Your last version is perfectly clear. If you need to use it more than once then define a short symbol the first time. That could  be as simple as $\sum_0^M f$.
Here are two suggestions each using just a single $\sum$.
$$
\sum_{x \in [1, 2, \ldots, M]^n} f(x)
$$
$$
\sum_{x_i = 0, \ (i = 1, \ldots n)}^M  f(x_i, x_2, \ldots, x_n)
$$
