$3$-adic expansion of $- \frac{9}{16}$ I get the $3$-adic expansion to be $1+1 \cdot 3+2 \cdot 3^2 +2 \cdot 3^3 + 0 \cdot 3^4+\cdots$. I'm trying to work out a pattern of the coefficients and think it is $1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 0, 0,...$. To show this I need to show that $1+1 \cdot 3+2 \cdot 3^2 +2 \cdot 3^3 + 0 \cdot 3^4 + \cdots$ sums to $- \frac{9}{16}$, but I don't know how to manipulate this series in order to compute it.
Any help would be appreciated!

As i understood, your question was how to find fraction form for
$...002211002211\in \mathbb Q_3$ (but not how fo find $p$-adic expansion for rational number).
Algorithm is the same, as for $p$-ary expression:
$$
...002211002211=2211(...+3^{12}+3^6+1)=\frac{54+18+3+1}{1-3^6}=-\frac{76}{728}=-\frac{19}{182}.
$$
