Summation notation not clear

Question

This must be a beginner's question, so I apologize in advance. I just could not find the answer anywhere.

I understand what the following means.

but what does this mean?

Is it simply the sum for all possible i's? Thank you.

Answer 1

If it's any comfort the fault is in the writer, not you.

Although the most common way to write a sum is as $\sum\limits_{i=a}^b a_i$ which means to add up all the $a_i$ as $i$ "runs through the integers $a$ to $b$, but you can really have any notation $\sum\limits_{i\text{ has any condition we specify}} a_i$ we mean to add up all the $a_i$ so that $i$ has the condition we want.

A cute example can by $\sum_{i|28} i$ would me to add all the integers (presumably positive) that divide $28$ so $\sum\limits_{i|28} i = 1 + 2+ 4+ 7+14 + 28=56$.

And as Don Thousand commented; perhaps the most common is that you may have some set $S= \{-7, 3,5, 49, -12, 15\}$ say then we can write $\sum\limits_{i\in S} 3^i$ to mean $3^{-7} + 3^3 + 3^5 + 3^{49} + 3^{-12} + 3^{15}$ (why we'd want to add those particular powers of $3$ is another question....)

So what does $\sum\limits_{i} 3^i$ mean? Well, I have no idea. I know that the sub "$i$" just means that "$i$" is the variable we base our summands. $\sum\limits_{i} 3^i$ means to add the powers of $3^i$ for ....some.... set of values of $i$. But which values of $i$? I can only assume the author had some in mind and thought it would be clear in context.

Answer 2

It's all about context. $\sum_n 3^n = \sum_{n \in S} 3^n $ for some implied set $S$. The writer must have mentioned the set being summed over. Usually $S=\{1,2,3\}$ or $S=\{1,2,3,...\}$ (though be careful since that notation only works for convergent series).

It's useful to write things that way when you're proof involves lots of sums, namely those summing over the same set.