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