Solution
= Solution
The weighted shift is compact because truncating after the first $N$ weights gives finite-rank operators converging in norm, as $1/n\to0$. Moreover
$$
\|R^m\|\leq\frac1{m!},
$$
so its spectral radius is zero. Since the <spectrum> is nonempty,
$$
\boxed{\sigma(R)=\{0\}.}
$$