So the textbook Sigma notation is one which has i=1 on the subscript and the n on the top of the sigma notation, whereas in some texts there is an i on the subscript and no "=1" following it nor an "n" on top of it.
In any case, do both mean the same thing? I.e. even without a =1 or an n, does a variable underneath the sigma sign mean that the units that are being summed vary from 1...to a finite (or infinite) n?
$\sum\limits_i~a_i$ where limits are not specified is a somewhat dirty way of notating that $i$ should range over all possible values of $i$. If it is well understood that $i$ can range over all positive integers, then this is equivalent to saying $\sum\limits_{i=1}^\infty a_i$
Otherwise you could write things such as $\sum\limits_{i\in\Delta} a_i$ where $i$ ranges over all elements in the indexing set $\Delta$. For instance, letting $\Delta = \{1,3,7\}$ you would have $\sum\limits_{i\in\Delta} a_i = a_1+a_3+a_7$. This notation is particular powerful as it allows us to talk about summing over things which don't commonly appear with an index ranging over natural numbers, for example summing over permutations such as $\sum\limits_{\sigma\in S_n} \sigma(1) = n!\cdot\frac{n+1}{2}$
If $i$ is sitting alone under the summation sign, its range has to be inferred from context.
Take this example: $$\sum_ix_i=v(N)$$ This came from one of my lectures on cooperative games. It is not immediately clear what $i$ ranges over. But the context gives that $i$ ranges over the set of players $N$, and $\vec x$ is an imputation (division of the payoff $v(N)$). The above equation is then the criterion for efficiency of the payout.
