Summary: THE TAYLOR POLYNOMIAL ERROR FORMULA
Let f(x) be a given function, and assume it has deriv-
atives around some point x = a (with as many deriva-
tives as we find necessary). For the error in the Taylor
polynomial pn(x), we have the formulas
f(x) - pn(x) =
(n + 1)!
(x - a)n+1f(n+1)(cx)
(x - t)nf(n+1)(t) dt
The point cx is restricted to the interval bounded by x
and a, and otherwise cx is unknown. We will use the
first form of this error formula, although the second
is more precise in that you do not need to deal with
the unknown point cx.