This is perhaps a daft question, but one that came up in discussion today and elicited different responses, so I thought I'd ask here. Imagine a function with the summation operator, like the arithmetic mean of some vector $x$. For this example, consider $x$ to be composed of an integer number of monotonically increasing elements, so that $x$ is effectively a list of values and not a function.
$$A = \frac{1}{n}\sum_{i = 1}^{n}x_{i}.$$
Now imagine you decided you were doing to remove / add elements in this vector, and wanted to specify how much the mean would change as $n$ changes. Naively, you might just write something like
$$\frac{\partial A}{\partial n} = \frac{-1}{n^2}\sum_{i = 1}^{n}x_{i}.$$
In this is anyway meaningful, or a completely wrong-headed mangling of what one can do with summations? I suspect it's pretty much useless given that the vector itself is not a function, but open to correction!