Let $K$ be a non-archimedean complete field and let $\pi \in K$ be an element with $0<|\pi|<1$. The Tate algebra $K\langle x, y\rangle $ is $$K\langle x, y \rangle := \left\{\sum\limits_{i, j \in \mathbb{N}} c_{ij}x^iy^j \mid \lim\limits_{i,j \to \infty} |c_{ij}|=0\right\}$$
Let $A=K\langle x, y\rangle/(y-\pi x)$. What do the elements of $A$ look like? Something I'm reading says
$$\sum\limits_{n\in \mathbb{N}} a_{n}x^n$$ converging for $|x| \leq \pi$, (so I guess $\lim |a_n\pi^n| \to 0$).
My attempt was: We have in $A$ (where $y=\pi x$) $$\sum c_{ij}x^iy^j = \sum c_{ij}\pi^jx^{i+j}=\sum\limits_{n \geq 0}\sum\limits_{i=0}^n c_{i,n-i}\pi^{n-i}x^n$$ so set $$a_n=\sum\limits_{i=0}^n c_{i,n-i}\pi^{n-i} $$ Then $$a_n\pi^n=\sum\limits_{i=0}^n c_{i,n-i}\pi^{2n-i} $$ but I got stuck.