There's many things that a 3rd quarter of calc
``Sequences \& Series''
can be about.

The key object is the Taylor series of a function.
This opens a whole can of worms, and suggests one ask:
- topology of function spaces
- approximation of functions
  (by poly, by trig poly, etc.)
- formal power series
- convergence of power series (over complex numbers)
  ...and Complex analysis d'apres Weierstra\ss

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Something I haven't discussed, and should:
over what -fields- do power series make sense?
-formally- (i.e., algebraically), they make sense over any field,
or indeed ring.
in order to represent functions, however, you need to work over a
complete field -- and experience with $\bR/\bC$ suggests that you want to work over
an algebraically closed one, though I'm not sure if this actually
happens over the $p$-adics...

Indeed, the ring of formal power series over $K$ is a local ring, with
maximal ideal $(x)$, and the Laurent series is its field of
fractions...
(hmmm....and power series in $(x-a)$ are the completion at the prime
$(x-a)$, right?)
...which can be used to introduce $p$-adics (they're like power
series! namely, they're a local ring...)

-----------------------------------------
I have -no- idea about the interest of looking at rational power
series over $\bC$ -and- the $p$-adics.

...though it's interesting to look at some series, like the geometric.

One interesting question though that this raises is the $p$-adic
valuation of $n!$ (this comes if you try looking at the radius of
convergence of $\exp$ over $p$-adics).
One sees that
$\abs{n!}_p &=
   \floor{\frac{n}{p}}
 + \floor{\frac{n}{p^2}}
 + \floor{\frac{n}{p^3}} + \dotsb\\
  &= \sum_{k=1}^\infty \floor{\frac{n}{p^k}}$
where this latter is always a finite sum.
In particular, it's 
 $ \leq \sum_{k=1}^\infty \frac{n}{p^k}
  =n\frac{1/p}{1-(1/p)}
  =n\frac{1}{p-1}$
(I bet one can do better, if you know some combinatorics)

Oh, and here's a power series that converges for all $\bC$ and
$p$-adics:
$a_n=[2^{n^2}3^{(n-1)^2}\dotsm p_k^{(n-k)^2}p_{n-1}]/q$
where $q$ is a prime such that $a_n<1/n!$
  (NB: $q\geq p_n$)
All the $p_k$ on top ensure convergence in $p_k$-adics, and the $q$ on
the bottom ensures convergence in $\bC$, without messing it up in the
$p$-adics (b/c it's just one big prime).
