I would argue that it is natural to introduce series first. Why are sequences interesting? The sequence $1$, $3/2$, $7/4$, etc. converges to $2$. Who cares? 

I think the most natural answer to ``who cares'' is series. Write $e = 1 + 1 + 1/2 + 1/6 + \cdots$ on the blackboard, and I expect that students will know what is meant, and think it's cool. We write $1/3 = .3333\dots$ in precalculus courses without first discussing convergence, and this isn't really all that different.

Having introduced series, one can continue and write things like $1 - 1 + 1 - 1 + \cdots$ or whatever on the blackboard, and perhaps scare the students a little bit and explain that it is possible to write down formulas which are complete nonsense. (Or maybe only almost-complete nonsense, Ramanujan argued in cold blood that $1 + 2 + 3 + 4 + \cdots = -1/12$.) This motivates a more cautious approach to the subject, i.e. discussing convergence of sequences.