Are these formulas proper notation? Are these valid ways of expressing the same idea? $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots = \sum_{x \in \mathbb N} \frac{1}{2^x} = \sum_{x = 1 }^ \infty \frac{1}{2^x} = \sum_{x = 1 }^ {x\to\infty} \frac{1}{2^x} = 1$$ The one that bothers me is the third one, it seems to imply that $x=\infty$ at some point, in other words it treats $\infty$ as a number. The problem with 1,2 and 4 would be that the sum only approaches 1 but never actually gets there. Yet, if you squint a little, they all look good.

In my opinion the third one is the correct one.

The first one is not exhaustive: you imagine it will go on with $\frac{1}{16}$, but it may as well be of the form $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+0+0+\cdots$.

In the second one you are assuming $0\
otin \mathbb{N}$, which is not exactly standard, as Peano axioms put $0$ in $\mathbb{N}$.

For the fouth one, the precise form could be $$\lim_{y\to\infty}\sum_{x=1}^y\frac{1}{2^x},$$ which actually is the definition of $\sum_{x=1}^{\infty}\frac{1}{2^x}$.