Understanding $p$-adic series normalization.

Question

I don't quite understand $p$-adic series normalization. I know that a $p$-adic series $\eta\in\mathbb{Q}_p$ is normalized if it satisfies: $$\eta=\sum_{i\in\mathbb{N}_{>\nu}}a_ip^i$$ Where $a_i\in\mathbb{Z}_p$ belongs to the integers modulo $p$, and $\nu\in\mathbb{N}$ is any starting index. Now, a general series $\tau\in\mathbb{Q}_p$ is defined by: $$\tau=\sum_{i\in\mathbb{N}_{>\nu}}r_ip^i$$ Where $r_i\in\mathbb{Q}$ is a rational number $r_i=\frac{a_i}{b_i}$ with numerator $\mathbf{n}(r_i)=a_i$ and denominator $\mathbf{d}(r_i)=b_i\nmid p$. Suppose i want to find the normalized form $\tau_N$ of $\tau$. I think i can define it to be: $$\tau_N=\frac{\sum_{i\in\mathbb{N}_{>\nu}}\mathbf{n}(r_i)p^i}{\prod_{i\in\mathbb{N}}\mathbf{d}(r_i)}$$ provided $0≠\mathbf{d}(\tau_N)≠\infty$. Is this a good approach? Is there another closed form for $\tau_N$ like the one i tried to describe?