I read somewhere that if the p-adic expansion of a p-adic number has repeating digits from some point on if and only if the p-adic number is a rational number.
I wanted to test this with the 2-adic number $a=1+2+4+8+16+\cdots$. According to the above, $a\in \mathbb{Q}$ and we have \begin{array}{rcl}a=\dfrac{1}{1-2}=-\dfrac{1}{2}.\end{array}
However, when I multiply this with the 2-adic number $-2$ I don't get $1$. Did I do something wrong?