Sum (sigma) notation disparity

Question

this might be a stupid question but I've googled it and I can't find an answer that I "trust". I am dealing with PCA (I guess that's not relevant, but just in case it is) and I am seeing a lot of Sigma notation that I'm not used to. My whole life I've always seen the following:

$\sum_{i=1}^{10}$

But now I am seeing this:

$\sum_{i}$

Does it mean exactly the same, supposing it is known that n=10 ??

Answer 1

When the range of the summation index is obvious/known from context,

$$\sum_i$$ has meaning. When the summation index itself is unambiguous, even

$$\sum$$ can do.

Answer 2

Basically, yes. It means that the index you are summing over is $i$, and presumably the domain for $i$ is stated elsewhere.

You might also see things like

$$ \sum_{a\in A} $$

or

$$ \int_A $$

etc.

Answer 3

A common notation for $$\sum_{n=1}^{\infty}a_n$$ is $$\sum_{n\in\Bbb N} a_n$$ We take it that every $n$ is to be summed, as $\sum_i a_i$ implies we sum across every value that $i$ is defined to take