\sum_{i\in\omega}(\omega+i)\cdot\omega^{i+2} = \\ =\sup_{n\in\omega}\sum_{i\in n}(\omega \cdot i) \cdot \omega^{i+2} = \\ =\sup_{n\in\omega} ((\omega\cdot 0)\cdot\omega^{0+2} + (\omega\cdot 1)\cdot\omega^{1+2} + (\omega\cdot 2)\cdot\omega^{2+2} + \dots + (\omega\cdot (n-1))\cdot\omega^{n-1+2}) = \\ =\sup_{n\in\omega} (\omega^4 + \dots +\omega^{n+2}) = \\ =\sup_{n\in\omega} (\omega^{n+2}) = \\ = \omega^{\omega}