= Solution
The summands $u_n=n^4/2^n$ are positive. Apply the <ratio test>:
$$
\frac{u_{n+1}}{u_n}=\frac{(n+1)^4}{2n^4}=\frac12\left(1+\frac1n\right)^4\longrightarrow\frac12<1.
$$
Therefore
$$
\boxed{\sum_{n=1}^{\infty}\frac{n^4}{2^n}\text{ converges absolutely}.}
$$
For a direct comparison interpretation, the ratio is eventually at most any fixed number between $1/2$ and $1$, such as $3/4$. The tail is then bounded by a convergent <geometric series>. Thus the exponential denominator wins over the fourth-power numerator strongly enough to make the whole <series> summable.
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