The classic picture for a Taylor series is to draw a function
(generally sin, though $\exp$ and $\log(1+x)$ are also good ones)
and the first few taylor polynomials -on the same picture-

This is okay, but it's cramped and dense:
a good alternative (better yet, supplement: I'd say do this first,
then the cramped picture)
is to draw a sequence of pictures
(following Edward Tufte's advice)
of successive Taylor approximations.
The idea is that a sequence of pictures -implies- motion:
if you see a few terms in a sequence, people can generally "fill in"
the rest of the sequence.
(note: here I mean "extrapolate". there's also interpolation, like you
do in a comic strip "between the panels", but that's not so relevant
for sequences)
...and if you put them -side by side-, it visually -shows- that it's a
sequence, and implies continuation, while piling them on top of each
other makes a mess.

[OTOH, note that there is -some- value in stacking 'em on top of each
 other: it lets you see that it converges to the right values.
 However, see notes on converge vs. asymptotic.]
