Notation in infinite sums and infinite unions, some questions Yesterday I see this expression:
$$(a, b)= \bigcup_{n=1}^\infty [a + 1/n, b - 1/n]$$
The first thing that I thought was "why this expression represent an open interval and not a closed one?" Then I suppose that this is because the above expression doesn't represent a limit, if it would be a limit then it would represent a closed interval instead of an open one.
I dont know if my reasoning is ok or not, and this is one of the reasons to ask in mathexchange.
A) Then my first question is: it is ok my reasoning or it is not correct? And if it is not correct please, can someone explain to me why the infinite union doesn't represent a closed interval? Because is clear that $1/n$ when $n=\infty$ is zero.
B) Now the second question is, there is an abuse of notation when someone write $\sum_{k=0}^{\infty}f(k)=c$ where $c\in\Bbb R$? I assume, because I read in some books, that a infinite summation represent a limit of a partial sum, so it makes sense that converges to a real number (or complex).
C) And the final question is, maybe 
$$(a, b)= \bigcup_{n\in\Bbb N} [a + 1/n, b - 1/n]$$
is a better expression than the original?. Thank you in advance.
 A: ""why this expression represent an open interval and not a closed one?""
Because that particular union of closed intervals is an open interval.  Nothing to do with representations or limits or anything.  The union simply is an open interval*.
It's easy to show that i) for $x \le a$ or $x \ge b$ then $x \notin$ any $[a + 1/n, b -1 /n] $ and ii) for $a < x < b$ then $x \in [a + 1/n, b - 1/n]$ for some $n \in N$.  So the union does in every sense of the word equal (a, b).
B) Infinite sums are limits of sequences of finite partial sums.  That would be an abuse of notation except the notation was specifically defined to mean a limit so you can't abuse a definition. 
But a sum is an entirely different thing than a union.  You can have infinite unions and I don't think you can have limits of unions.  You can't have infinite sums and you can have limits of sums.
C)  Your notation and the original notation both mean exactly the same thing and both are used and can be used interchangeably.
* It is worth noting all finite unions of closed sets are closed.  But not necessarily all infinite unions are closed as this example clearly demonstrates.  However all intersections, finite or infinite are closed.
All unions of open sets, finite or infinite, are open.  All finite intersections of open sets are open, but not all infinite intersections of open sets are open.
