Calculus: Analytic Geometry and Calculus, with Vectors

12.5 Taylor formulas with remainders 633

Integrating by parts with

u=f'(t) dv=dt

du = f"(t) dt, v = - (x - t)

gives

or

f(x) = f(a) + [-f'(t)(x - t)la -+- I Zf"(t) (x 1t)dt

(12.52) f(x) = f(a) + la (x - a) +

J

f" (t) (x 1 t)dt.

aa

Another integration by parts with

u = f' (t) dv =(x 1 t)dt

du = f" (t) dt, v = - (x - t) 2

2!

gives

f(x) = f(a) (x - a) (x - a)2 + / z f( )(t) (x2it) dt.

One more integration by parts gives the result of putting n = 3 in the

formula

(12.53) f(x) = f(a) + -i-a) (x - a) + f-2 a) (x - a)2

(") a

+. .. +f

n

(x - a)" + Rn(x),

where

(12.54) Rn(x) = 1 f t)^ dt,

and further integrations by parts give the results when n = 4, 5, 6,.

The formula (12.53) is a Taylor formula with remainder R,(x). The right

member of (12.54) is the integral form of the remainder. In some cases,

(12.54) and other remainder formulas can be used to determine values of

x for which

(12.55) lim R (x) = 0.

n-»

For such values of x the Taylor formula

(12.56) f(x) = f(a) (x - a) +f' (a) (x - a)2 +