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