[Can we get Chebfun into Maxima?]
If you're not aware of the "Chebfun" package developed by Trefethen, et al, at Oxford U., you will be pleasantly surprised.
Basically, Chebfun approximates smooth curves over a finite interval to within machine precision. Chebfun also supports piecewise smooth curves by separately Chebbing each piece.
Since different smooth curves require different numbers of terms to converge, some of these sums can have 1000 or more terms.
As pointed out in the slides & video below, the whole point of this is to provide the flavor of symbolic computation with the efficiency and ease of numeric computation. For example, plotting, derivatives, integrals, root-finding are all relatively easy in this framework.
The major success of Chebfun's is in ODE's, both linear & non-linear.
It's a pity that Chebfun only works on Matlab; perhaps some student could be motivated to do this in Maxima.
After watching the video below, I started to wonder why we bother teaching Taylor series at all.
Here's a 1-hour tutorial about Chebfun (850 MBytes):
http://downloads.sms.cam.ac.uk/1160831/1160835.mp4
Here are the slides for this talk (1 MByte):
http://people.maths.ox.ac.uk/trefethen/chebfun.pdf
Some history of Cheb approximations
http://people.maths.ox.ac.uk/trefethen/cftalk.pdf