TITLE: Is it bad to call series a generalization of sum?
QUESTION [6 upvotes]: In a recent question I asked why series has a name separate from that of sum, and the general answer was that a series does not have the nice properties of sum.
Does this mean it is bad to call series a generalization of sum?

REPLY [9 votes]: Concept $X$ generalizes concept $Y$ if every instance of $Y$ is a special case of $X$, at least with regard to some relevant property of the concepts involved.
The infinite series $\sum_{n=1}^{\infty}s_{n}$ is defined to be the limit of the partial sums of a sequence $(s_{n})$.
Given a "finite summation", it can be thought of as an infinite series where, for some $N$, we have $s_{n}=0$ whenever $n>N$, in the sense that the infinite series will converge to the value of the finite summation (this is the "relevant property" in this case).
Therefore the concept of an infinite series can be regarded as a valid generalization of the usual concept of summation, so the answer to your original question is no.