I don't quite follow Spivak's derivation of Taylor's theorem;
doing it via integration by parts is very straightforward
...is it as general?
(or is Spivak just being too clever by half in his proof?)

There's at least 3 useful forms of the error term:
Integral form
  (should this be called "Taylor's form"?)
Cauchy
Lagrange

Spivak proves Cauchy; I can prove Integral
The Lagrange follows from Cauchy by applying Cauchy MVT
(no idea about why Cauchy didn't see that ;-)

The Lagrange form of the error is the only one to memorize, if you
must memorize one, as it's short.
(however, it's not true over $\bC$, though the other forms work fine)

However, you shouldn't bother:
just note that it's something that depends on:
- the next derivative
- the distance from $a$, raised to the corresponding power
- divided by a factorial!

thus if the derivatives don't blow up (faster than a factorial),
the series converges, and does so -uniformly- on a compact set.

This theoretical consequence is the most important;
the rest is computational.
