520 Chapter 8 • Infinite Series of Real Numbers
c and x, where x -:/:- c, and f(n+l)(t) exists for all t in the open interval I
between c and x. With Tn(x) and Rn(x) as defined above,
(a) 3 zEI3
f(n+l)( )
R (x) = z (x - c)n+l.
n (n+ 1)!
(29)[Formula (29) is called the "Lagrange form" of the remainder.]
(b) if f(n+l) exists and is integrable on [c, x] if c < x, or [x, c] if x < c,
Rn(x) =I^1 ix (x - t)nf(n+l)(t)dt.
n. c(30)[Formula (30) is called the "integral form" of the remainder.]
(c) if f(n+l) is continuous on I= [c, x ] if c < x, or I= [x, c] if x < c, then
3z EI 3
f(n+l)(z)
Rn(x) = 1 (x - z)n(x - c). (31)
n.[Formula (31) is called "Cauchy's form" of the remainder.]Proof. Part (a) was proved as Theorem 6.5.11 and Part (b) as Theorem
7.6.16. We can derive (c) from (b) by applying the first mean value theorem for
integrals (Theorem 7.6.17) to (b). See Exercise 1. •
We shall now show by means of examples how Taylor's theorem, with its
various forms of the remainder, can be used to show that some familiar functions
are analytic at a given real number c.
Examples 8. 7.4 (Some Analytic Functions^9 and Their Maclaurin
Series)00 k
(a) ex= L ~! (valid for all x E JR).
k=O. oo (-l)kx2k+1
(b) smx = L ( 2 k + l)! (valid for all x E JR).
k=O
oo (-l)kx2k
(c) cosx = ~ ( 2 k)! (valid for all x E JR).
9. For a discussion of the problem of defining these functions, see Section 7.7.