Is summing from $i=1$ to $\infty$ the same as summing over $\mathbb N$? Are the series $$\sum_{i=1}^\infty f(i)~~~~~~~~~~~~~~~~~~~~~~~~~~~\sum_{i\in\mathbb N}f(i)$$
identical? The first one adds $f(1)+f(2)+\cdots$, and so does the second one.
 A: Not necessarily, though maybe.
The first notation has a precise definition as limit:
$$\sum_{i=1}^\infty a_i:=\lim_{n\to\infty}\sum_{i=0}^na_i.$$
On the other hand, how can we define $$\sum_{i\in I}a_i$$
for an arbitrary index set $I$ in the first place? For finite $I$ this is easy: If $I=\emptyset$, we define the set to be $=0$, and otherwise we pick $i_0\in I$ and can define recursively
$$\sum_{i\in I}a_i:=a_{i_0}+\sum_{i\in I\setminus\{i_0\}}a_i, $$
which is well-defined by the laws of commutativity and associativity. 
This does not make any assumptions about any structure (such as: an order relation) of $I$. 
For infinite sets, to be on the safe side  we can define the sum if all but finitely many summands are zero, i.e., 
$$ \sum_{i\in I}a_i:=\sum_{i\in J}a_i$$
if $J$ is a finite set and $a_i=0$ for all $i\in I\setminus J$. Beyond that, we no longer have a sum, we have a series. 
For the special case that $a_i\ge 0$ for all $i\in I$, it might make sense to define
$$\sum_{i\in I}a_i:=\sup\Bigl\{\, \sum_{i\in J}a_i\Bigm|J\subseteq I, |J|<\infty\,\Bigr\}$$
But beyond that, recall that for series with countably infinite index set, the order of summation may matter (the distinction of conditionally convergent vs. absolutely convergent). Hence we cannot for example define $\sum_{i\in \Bbb Q}a_i$ as $\sum_{i=0}^\infty a_{q(i)}$ where $q\colon \Bbb N\to \Bbb Q$ is an enumeration of the rationals: There is no canonical such enumeration, hence if this produces a conditionally convergent series, the result is not well-defined. For the special case $I=\Bbb N$ (or even some slightly more general cases) one may argue that there is a canonical enumeration of $I$ and thus define $\sum_{i\in\Bbb N}a_i$ as $\sum_{i=0}^\infty a_i$. But that at least this may mean an inconsistency of notation if one uses $\sum_{i\in I}$ for some other infinite index sets as well.
