I think, from the algebraic point of view, that this doesn't make sense, since the addition of $n$ terms is defined "inductively" from the binary operation of the addition. This even yields the formula of $nx = x + x + \cdots + x$, where the $x$ is added $n$ times (which I saw as a different operation than the multiplication in a ring, but an action of the natural numbers in the ring).
However, it would make sense to make a definition of adding $n \in \mathbb{R}\setminus\mathbb{N}$ terms or summands? Something like the sense of a fractional derivative?
Markus Muller and Dierk Schleicher say there is.
http://www.mpmueller.net/HowToAdd.pdf