Calculus: Analytic Geometry and Calculus, with Vectors

12.5 Taylor formulas with remainders 635

and use the integral form (12.54) to obtain

(12.581) R..(x) =q(q - 1)(q - 2)...

(q-n)

n!

fx

(1 +t)4-1 (---:)' A

The function 0 for which 4,(t) = (x - t)/(1 + tt)is monotone over the

interval from 0 to x and O(x) = 0, so j4(t)( must attain its maximum

over the interval from 0 to x when t = 0. This maximum is therefore

jxi. Hence

(12.582) IR,.(x)l < jq(q - 1)(q - 2)

. (q - n)l

1xI" fox (1 + )q-1 dt

In case q is a nonnegative integer or x = 0, it is easy to see that

(12.583) lim R (x) = 0,

because Rn(x) = 0 for each sufficiently great n. When q is not a non-

negative integer and 0 < jxj < 1, an application of the ratio test gives

(12.583). This establishes the binomial formula (12.58) for the case

in which lxj < 1.

Problems 12.59

1 With the aid of Taylor formulas with remainders, obtain the expansions

off in powers of x - a when

(a) f (x) = ex, a = 0 (b) f (x) = e=, a = 1

(c) f(x) = sin x, a = 0 (d) f(x) = sin x, a =

(e) f(x) = cos x, a = 0 (f) f(x)=cosx,a=

2 Supposing that jxi < 1, write two more terms in each series appearing in
the calculations

sin-' x = fox 11-

t?

dt

i i s

(1-x)-3j=1+ (-x)+i

1-3=1r+ix+2.4xa+ ...

1 1-3

sin-' xOx [1+Zt2+2.4t4+ +

sin-,x=x+213xs+.2145xs+214.657x7+ ...