Infinite Ordinal Sum

Ordinal summation requires an ordinal index. And $\infty$ is not an ordinal.

Other than that, the summation does make sense in general. If $I$ is a linearly ordered set, and $x_i$ is a linearly ordered set for each $i\in I$, then $\sum_{i\in I}x_i$ would be the order type obtained by replacing $i$ with $x_i$, and considering the "[somewhat-]lexicographic order" obtained.

If $I$ is an ordinal and each $x_i$ is an ordinal, it turns out that the sum is an ordinal as well. Which is why everything works out.

As far as notation goes, I'd probably go for $\sum_{i<\alpha}$ and not $\sum_{i=1}^\alpha$. Which will also allow you to catch those pesky limit cases.