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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 canserve 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 iffor 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 numbersthen 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 convergenceof series are quite independent from that of sequences. And it does not sound impossible to do series prior to sequences.