You also asked for references. I have no doubt that you can find this in many books. (Quite often they are left to exercises.) Using Google Books Search I was able to find a few references in textbooks.
When I searched Google Books for sum ordinals disjoint I found in the book Stoll R. Set theory and logic (Dover 1963, 1979) on p.313
The concept of the ordinal sum of two well-ordered sets extends directly to an arbitrary (well-ordered) family of well-ordered sets. First, a word about the notation for such families. In view of Lemma 7.6 we may take the indexing set to be an ordinal number. We shall do this and use notation like $\{a_\xi; \xi\in\eta\}$ for such a family. If, then, $\{a_\xi; \xi\in\eta\}$ is a disjoint family of well-ordered sets, indexed by (the ordinal number) $\lambda$, we define its ordinal sum as $\bigcup_\xi a_\xi$ ordered as follows: If $x$ and $y$ are members of the union and in the same set $a_\xi$ then the order in $a_\xi$ prevails; if $x\in a_\xi$ and $y\in a_\eta$ where $\xi<\eta$, we take $x < y$.
When I searched Google Books for sum ordinals transfinite sequence I found that in the book Peter Komjath, Vilmos Totik: Problems and Theorems in Classical Set Theory these notions are used and they are defined more generally for order types on p.32:
If $\theta_i$ are order types for each $i\in I$ where $\langle I,<\rangle$ is an ordered set, then we define $\sum_{i\in I} \theta_i$ to be the order type of the ordered union of $\langle A_i, \prec_i\rangle$, $i\in I$, with respect to $\langle I,<\rangle$, where $\langle A_i, \prec_i\rangle$ are pairwise disjoint ordered sets with ordered type $\theta_i$.
Ordered union is defined on p.24:
Let $\langle A_i,<_i\rangle$, $i\in I$ be ordered sets with pairwise disjoint ground sets $A_i$ and let the index set $I$ be also ordered by the relation $<$. The ordered union of $\langle A_i,<_i\rangle$, $i\in I$ with respect to the ordered set $\langle I,<\rangle$ is the ordered set $\langle B,\prec\rangle$ in which $B = \bigcup_{i\in I}A_i$, and for $a \in A_i$ and $b \in A_j$ the relation $a < b$ holds if and only if $i < j$ or $i = j$ and $a <_i b$.