Solution
= Solution
For $u_n=2^{-n}n^k$,
$$
\frac{u_{n+1}}{u_n}=\frac12\left(1+\frac1n\right)^k\longrightarrow\frac12.
$$
Thus for all sufficiently large $n$ the ratio is, for example, at most $3/4$. Comparison with a decaying <geometric sequence>, equivalently the <ratio test>, yields
$$
\boxed{\lim_{n\to\infty}2^{-n}n^k=0}.
$$