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 £© Section
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 £© Taylor’s Theorem
The goal of this section is to prove that if Pn is the nth order Taylor polynomial for a
function f at a point c, then, under suitable conditions, the remainder function
Rn(h) = f(c + h) − T (c + h) ()
is O(hn+1). This result is a consequence of Taylor’s theorem, which we now state and
prove.
Taylor’s Theorem Suppose f is continuous on the closed interval [a, b] and has n + 1
continuous derivatives on the open interval (a, b). If x and c are points in (a, b), then
f 00(c) f (n)(c)
f(x) = f(c) + f 0(c)(x − c) + (x − c)2 + · · · + (x − c)n + r (x), ()
2! n! n
where
x − n
(x t) (n+1)
rn(x) = Z f (t)dt. ()
c n!
That is, if Pn is the nth order Taylor polynomial for f at some point c in (a, b) and x is
any point in (a, b), then
f(x) = Pn(x) + rn(x), ()
where rn is given by ().
We will show that Taylor’s theorem follows from the Fundamental Theorem of Integral
bined with repeated applications of integration by parts. Let f be a func-
tion satisfying the conditions of the theorem. Since f is an antiderivative of f 0, by the
Fundamental Theorem of Integral Calculus we have
x
0
f(x) − f(c) = Z f (t)dt. ()
c
Hence x
0
f(x) = f(c) + Z f (t)dt, ()
c
which is the statement of Taylor’s theorem when