_Caveat: below I am only saying that it could be done, not that it should. I am in the nets/sequences first camp._

I think implicit in the sentiment behind [this comment](https://mathoverflow.net/questions/105509/another-chicken-or-egg-sequence-or-series#comment271755_105509) is 

> How can you define convergence of a series without referring to the convergence of its sequence of partial sums. 

Because [as we know](https://mathoverflow.net/questions/105509/another-chicken-or-egg-sequence-or-series#comment1009465_105509), sequences are [telescoping series](https://en.wikipedia.org/wiki/Telescoping_series) are practically indistinguishable. 

And I think in principle this *could* be done, and in some way can be a good motivation for the notion of completeness. 

Essentially, one can start by motivating the idea that a series converges (settles down to a number) if its tail becomes "negligible", meaning that all the changes past a certain point is very small. This one can expand to meaning "past a certain point, the sum of any consecutive string of numbers in the series is no more than a fixed error". (Basically start by motivating, the definition for the series being Cauchy.) 

Then one can start asking the question of whether there actually _is_ a number that is being represented by this infinite sum (Cauchy completeness). 

--------------

BTW, thinking of a sequence as a telescoping series is occasionally useful in analysis (most textbook proofs of Banach fixed point, for example).