TITLE: Is $ \sum_{x=1}^{\infty} \frac{1}{2^x} = 1 $ an abuse of notation?
QUESTION [5 upvotes]: Is $ \sum_{x=1}^{\infty} \frac{1}{2^x} = 1 $ correct or is it just a shortened way to say $ \lim_{y \to \infty } \sum_{x=1}^{y+1} \frac{1}{2^x} = 1 $ ?
In a softer way of asking this question if we could add up all the numbers of the sequence would it equal 1 or are we treating the limit of the partial sums as the sum as a matter of convenience?
It was a something I was wondering about when I was recalling the classic joke "$\aleph_0$ mathematicians walk into the bar the first orders half a glass of beer..." and thinking a little philosophically about if the sum was "really" the sum of the cups or something else.

REPLY [15 votes]: Recall that $\sum\limits_{k=1}^{\infty}x_k$ is, by definition, $\lim\limits_{n \rightarrow \infty} s_n$, where, for every $n\geqslant1$, $s_n=x_1+\cdots+x_n$. There is no abuse of notation. There may be, perhaps, a misinterpretation of notation, taking the notation to mean more than it describes, but that is on the reader's behalf.