412 Chapter 7 • The Riemann Integral
a and x, where x -=/=-a. Suppose f(n+I) (t) exists and is integrable on [a, x ] if
a < x, or [x, a] if x < a. Let Tn(x) denote the nth Taylor polynomial for f
about a, and R,,,(x) denote the "remainder" f(x) - Tn(x). Then,
(28)Proof. Suppose f, x, a, and I satisfy the above hypotheses. The proof will
proceed by mathematical induction. You will be asked to fill in the missing
details in Exercise 15. Vn E N , let P(n) denote the statement that Equation
(28) is true.
Part 1 (Prove P(l)): We apply integration by parts to the right side of
Equation (28) with n = l. Letting u = x-t and dv = f"(t)dt, we have du= -dt
and v = f' ( t), so integration by parts yields
~ 1x (x - t)J" (t)dt = :! { [(x - t)f' (t)J: -1x f' (t)(-dt)}
(fill in details)= f ( x) - Ti ( x)
= Ri(x).Part 2 (Prove P(k) =? P(k + 1)): Assume P(k). That is,
We apply integration by parts to the right side of Equation (28) with
n = k + l. Letting u = (x - t)k+^1 and dv = f(k+^2 l(t)dt, we have du= (k +
l)(x - t)k(-dt) and v = f(k+ll(t), so integration by parts yields
1 11 x(x-t)k+lf(k+2)(t)dt= 1 ,{[(x-t)k+Ij(k+I)(t)]x
(k+l). a (k+l). a-1x j(k+I)(t)(k + l)(x - t)k(-dt)}
(fill in details)f(k+Il(a)
= - (k + l)! (x - a)k+I + Rk(x)