When Should you use: $\sum_ix_i$, over: $\sum_{i=1}^Nx_i$? I seem to see the former notation quite regularly in physics. My question is, is the former just sloppy summation notation? Or is actually preferable to use it at some point?
For example, say I have $N$ values represented by $(x_1, x_2, \dots, x_i, \dots, x_n)$ where $x_i$ represents any possible value, is it ever correct to write:
$$S = \sum_ix_i$$
To mean: "The sum of all possible values $x_i$", or is it always preferable to write:
$$S = \sum_i^Nx_i$$
What about the case where I don't actually know the total number of values $N$? I realise I could take $N$ to be some constant which I don't necessarily have to define, but in that case would the former notation be acceptable?
 A: Context is key. At the end of the day, notation has to be understandable. If it is clear that the values of $i$ come from the set $\{1,\ldots,N\}$, then it is usually fine to just state $i$ on the bottom of the summation with the understanding that the reader should interpret that as "all possible values of $i$ from the set in which it lives", which is dependant on context. Putting $N$ on the top of the summation is more explicit about this, and should definitely be used if there is any ambiguity about when you want the summation to stop.
A: Typically one would write $\sum_{i=1}^Nx_i$ in mathematics. This allows for a lower bound of the sum beginning with a number not equal to $1$. It can be preferable to use alternative notation when the indices $i$ are not consecutive integers. For example, if $C$ is the set of countries, and each $c\in C$ assigned a population $p_c$, then world population (pretend no one has dual citizenship) is $\sum_{c\in C}p_c$. We might be able to assign each $c$ an integer, and express the sum over integers, but this is unnatural and pointless.
