This is a side question which is more motivated by teaching than research.
First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" quantities; on the other hand, decimal expansions — especially infinite — are more likely to be series).
Secondly, is it natural for sequences to be placed prior to series in a calculus course?
So, which one is more original, a sequence or a series?
After-dinner edit. We define a sequence to be… a function mapping the positive numbers to a set(?). We define a series to be…
a formal infinite sum $\sum_{n=1}^\infty a_n$(?). Tell me what is your way to "define" these two guys, I do not believe they are very related.
There are no doubts that it is easier to define convergence of series via convergence of sequences, but it does not imply their "primogeniture".
The notion of Cauchy sequence is an elegant way to build the apparatus of not only sequences but also of real numbers; as such it can
serve as a definition of series: a series is a formal infinite sum $\sum_{n=1}^\infty a_n$, and it is called a convergent series if
for any $\epsilon>0$ there exists an $N=N(\epsilon)$ such that for any $m>n>N$ the sum $|a_n+\dots+a_m|<\epsilon$. The real numbers
then are nothing but representatives of equivalence classes of convergent series. (I have no desire here to expand all the details.)
A sequence $b_n$ is convergent when the corresponding series $\sum_{n=1}^\infty a_n$ where $a_1=b_1$ and $a_n=b_n-b_{n-1}$ for $n\ge2$
converges. It would be honest to say that, besides the trivialities like "algebra of limits", the techniques for investigating convergence
of series are quite independent from that of sequences. And it does not sound impossible to do series prior to sequences.
Historically, all these convergence/divergence issues were purely intuitive for both sequences and series, and they both were on the market for many
centuries. I ask whether their exists an overwhelming historical support to the notion of sequence to lead.
Best Answer
This is not a precise answer, mainly some thoughts.
Historically both ideas seem very old (2000+ years), and sometimes it is hard to tell what point of view is predominant.
Say if one looks at the Method of Exhaustion or things around Zeno's paradoxes what does arise 'sequences' or 'series'? For some of the constructions, the one seems more natural for others the other. In any case, in some form series already arose then.
One more point in favor of the fact that series where around early on: while in today's courses differentiation comes before integeration, in an intuitive sense I think integration is rather the easier or more natural idea, and historically early forms of integeration are very old (cf. Method of Exhaustion, which in my opinion counts as some sort of integeration). And integration and series sort of go together.
For your second question, for a course today, I would however start with sequence.
If one wants to talk about series in a somewhat rigorous form just having the notion/word sequence at ones disposal already seems like a big plus.