\begin{aligned}
 \sum_{i\in\omega}\big((\omega+i)^3+3^{\omega+i}\big)
 & = \sum_{i\in\omega}\big(\omega^3+\omega^2\cdot i + \omega\cdot i + i + \omega\cdot 3^i\big) \\
 & = \sup_{n\in\omega}\sum_{i\in n}\big(\omega^3+\omega^2\cdot i + \omega\cdot (i + 3^i)\big) \\
 & = \sup_{n\in\omega}\big(\omega^3\cdot n + \omega^2\cdot (n-1)+\omega\cdot (n-1+3^{n-1})\big) \\
 & = \omega^4
\end{aligned}