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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?

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    $\begingroup$ No, generalizations don't have to have all the nice properties of the thing they are generalizing. $\endgroup$ – Rahul May 21 '16 at 21:31
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    $\begingroup$ Clearly the intent of infinite sums is to be ... infinite... sums. Much of the time, everything is fine. Sometimes something's wrong. Not such a big deal, and certainly not a good reason to hijack terminology that suggests its own meaning (as opposed to artificial nomenclature made for no other reason than to avoid unwarranted intuitive leaps. ... and is that avoidance even the primary goal? Isn't our first guess by intuition based on admittedly too-naive previous examples?! Of course it is.) $\endgroup$ – paul garrett May 21 '16 at 21:34
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    $\begingroup$ Could you please elaborate what you mean by "bad", or specify in what context(s) you want to call a "(convergent?) series a generalization of a sum"? (This isn't rhetorical nitpicking. Without knowing your intent, it's difficult to answer other than by opinion.) $\endgroup$ – Andrew D. Hwang May 21 '16 at 21:57
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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.

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  • $\begingroup$ Definitely, there is no other answer. Series are the standard generalization of a finite sum in classical analysis, so how can it be bad to call them what they are? Sure they don't share all the properties of the sum; if they did, they would be operationally indistinguishable, something you cannot demand of a generalization! $\endgroup$ – guest May 22 '16 at 10:29
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General topological spaces do not have a large number of the nice properties that metric spaces have. E.g. without the Hausdorff property a sequence in a topological space can have more than one limit (a very undesirable property). I don't think, however, that anyone would disagree with the statement that topological spaces generalise metric spaces.

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