I find the use of sigma notation quite arbitrary. Sometimes the counters and limits are defined, sometimes they aren't. And sometimes, I just can't comprehend what it means.
For example, in my textbook, they've given an example,
$$\alpha^3 + p\alpha + q = 0$$ $$\beta^3 + p\beta + q = 0$$ $$\gamma^3 + p\gamma + q = 0$$
Now, here's the confusing part, they jump straight to saying:
$$\implies \sum \alpha^3 + p\left(\sum \alpha\right) + 3q = 0$$
which I can't make the head or tail of.
Without seeing the context, which might make things better, I'm inclined to blame the textbook here for being confusing. However, if they wrote:
$$\alpha_1^3 + p\alpha_1 + q = 0$$ $$\alpha_2^3 + p\alpha_2 + q = 0$$ $$\alpha_3^3 + p\alpha_3 + q = 0$$
then you could conclude that:
$$\sum_{i=1}^3\alpha_i^3+p\left(\sum_{i=1}^3\alpha_i\right)+3q=0$$
which should be clear (I hope). However, in this case, I would be happy enough with:
$$\sum\alpha_i^3+p\left(\sum\alpha_i\right)+3q=0$$
as I consider it fairly obvious that the sum must be over $i$, and the range should also be clear, so I can write a slightly less cluttered expression. This is a matter of taste though, and if I thought there was a chance of confusion I would take the first option.
The book seems to be assuming it's obvious that the sum is over Greek letters, and this is not a convention I support.
