= Solution
The shifted <cyclotomic polynomial> $\Phi_{p^{n+1}}(1+T)$ is Eisenstein at $p$: its constant coefficient is $p$, and modulo $p$ it is $T^{p^n(p-1)}$. Thus $[K_n:\mathbb Q_p]=p^n(p-1)$, $\zeta_n-1$ is a <uniformizer>, and $[K_n:K_{n-1}]=p$ for $n\geq1$. The $p$ automorphisms over $K_{n-1}$ send $\zeta_n$ to $\xi\zeta_n$ for $\xi\in\mu_p$.
Since $\zeta_n-1$ has positive valuation, evaluating an integral <formal power series> there converges. A <unit> series has <unit> constant coefficient and evaluates to a <unit>. Evaluate the <Coleman norm operator> identity at $T=\zeta_n-1$:
$$
\begin{aligned}
N_{K_n/K_{n-1}}\bigl(f(\zeta_n-1)\bigr)
&=\prod_{\xi^p=1}f(\xi\zeta_n-1)\\
&=(Nf)(\zeta_n^p-1)
=f(\zeta_{n-1}-1).
\end{aligned}
$$
The final equality uses norm fixation and the compatible choice of roots. Hence
$$
\boxed{N_{K_n/K_{n-1}}(u_n)=u_{n-1}\qquad(n\geq1).}
$$
In particular, norm fixation of a single <Coleman power series> produces a <norm-compatible sequence of local units>.
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