Let $K/\Bbb Q_p$ be a finite extension with integers $\mathcal{O}$ and $\mathcal{O}[[T]]$ the ring of formal power series in one variable over $\mathcal{O}$. Define a "shift operator" by
$$ \tau_n : \mathcal{O}[[T]] \to \mathcal{O}[[T]] : \sum_{i=0}^\infty a_i T^i \mapsto \sum_{i = n}^\infty a_i T^{i - n}.$$
Why is $\tau_n(T^n f(T)) = f(T)$ for all $f \in \mathcal{O}[[T]]$ ?
I'm definitely being silly but I can't see why this is the case; when I apply $\tau_n$ to
$$T^n\sum_{i=0}^\infty a_i T^i = \sum_{i = 0}^\infty a_iT^{i + n}$$
I end up with
$$\tau_n\left(\sum_{i = 0}^\infty a_i T^{i + n} \right) = \sum_{i = n}^\infty a_i T^i \neq f(T).$$