The Real And Complex Number Systems

every g k is never zero in c , c c, wherek 0, 1, 2. . . , n. So, we can apply
n 1 times L-Hospital Rule methoned in Supplement, and thus get
fx
lim xc gx lim f n1 x
xc
g n1 x
lim xc
lim xc
f n1 x f n1 c
g n1 x g n1 c
f n1 x f n1 c/x c
g n1 x g n1 c/x c
fn c
g n c since fn exists and g n exists 0 on a, b.
5.29 Show that the formula in Taylor’s theorem can also be written as follows:
n1
fx
k0
f
k
c
k!
x c k x cx x 1 n1
n 1!
f n x 1 ,
where x 1 is interior to the interval joining x and c. Let1 x x 1 /x c. Show that
0 1 and deduce the following form of the remainder term (due to Cauchy):
1 n1 x c n
f
n 1!
n x 1 c.
Hint. Take Gt t in the proof of Theorem 5.20.
Proof: Let
and note that
n1
Ft
k0
f
k
t
k!
x t k ,andGt t,
F t
fn t
x tn1
n 1!
then by Generalized Mean Value Theorem, wehave
Fx FcG x 1 Gx GcF x 1
which implies that
n1
fx
k0
f
k
c
k!
So,wehaveprovethat
where
x c k
fn x 1
n 1! x x 1 n
fn x 1 c
n 1!
n1
fx
k0
f
k
c
k!
x c n 1 n ,wherex 1 x 1 c.
x c k R n1 x,
R n1 x fn x 1 c
x c n 1 n ,wherex
n 1!
1 x 1 c
is called a Cauchy Remainder.
Supplement on some questions.
1. Let f be continuous on 0, 1 and differentiable on 0, 1. Suppose that f0 0and