Convergence of $\sum\limits_{n=0}^{\infty}\frac{n^3+n^4}{2^{n+1}}$ Does $\sum\limits_{n=0}^{\infty}\frac{n^3+n^4}{2^{n+1}}$ converges (absolute)?

I've already tried the Ratio test and the Root test test, but they failed both...
Hints:

(1) For $\;k\in\Bbb N\;$ :

$$a_n:=\frac{n^k}{2^n}\implies\frac{a_{n+1}}{a_n}=\frac{(n+1)^n}{2^{n+1}}\frac{2^n}{n^k}=\frac12\left(\frac{n+1}n\right)^k\xrightarrow[n\to\infty]{}\frac12&lt;1$$

(2) Comparison

$$\frac{n^3+n^4}{2^{n+1}}\le\frac{n^4}{2^n}\;\ldots$$
Hint:
For each c>0, there exists $n_0$ s.t. if $n&gt;n_0$ then $n&gt;c\log(n)$ (by calculate the limit of $log(n)/n$) , then for $c=10$
you gets $2^\frac{n}{2}&gt;n^5$, and by sum of geometric sequence you gets $n^5&gt;n^4+n^3$ then...

Now, you can use comparison.