This is perhaps beyond intro calc, but it's a fun thing to do with
series: 
Frobenius' method
to solve
differential equations.

Basically, assume that a differential equation has an analytic
solution, expand it as a power series, and get a recursive formula for
the coefficients of the solution.
Yes, this is the obvious thing to do -- Frobenius got the name though.
This is easy to do in the special cases of $y'=y$ and $y''=k^2y$, but
works much more generally:
Fuch's theorem states that if you expand about a smooth point or a
regular singular point, Frobenius' method will give 

At least do it for $\exp$ and $\cos$ and $\sin$; here the recurrence
is very easy.

For extra fun, have them compute the Bessel functions using the
Frobenius method.
