1549901369-Elements_of_Real_Analysis__Denlinger_

8. 7 Analytic Functions 525

Case 2 (a< 1): Let M = max{l, ll + x l°'-^1 }. Arguing as in Case 1 above

(Exercise 10) we can show that

which completes the proof. •

1

Example 8.7.11 Maclaurin series for v'l+X:

l+x

1 1 1 ·3 1 ·3·5 1·3·5·7

---= 1-- x + -x^2 - --x^3 + x^4 - · · · on (-1, l].

v'1+X 2 2·4 2·4·6 2·4·6· 8

Proof. This is a binomial (1 + x)-^1!^2. By Theorem 8.7.10 its Maclaurin
series is

(1 + x)-1/2 = f (-~)xk.

k=O k

(

• ~)- (-l)k1·3·5·····(2k-l).
  Now, when k 2: 1, k -
  2
  kk!. (Show details.)

~ (-l)kl. 3. 5 ..... (2k-l)
Thus, the Maclaurin series is 1 + L... kkl xk
k=l^2.
1 1 ·3 1 ·3·5 1 ·3·5·7 1 ·3·5·7· 9
= 1 - 2x + 222! x2 - ~x3 + 244! x4 - 255! x5 + .. ..

By Theorem 8.7.10 we know the radius of convergence is 1. The series di-
verges when x = -1 but converges when x = 1. (See Exercises 8.2.43 and
8.3.13.) Since (1 + x)-^1!^2 is continuous at x = 1, Abel's theorem 8.6. 19 guar-
antees that it converges to (1 + x) -^1!^2 everywhere on the interval (-1, l]. 0

Example 8.7.12 Maclaurin series for sin-^1 x :

lx^3 1·3x^5 1 ·3·5x^7 1·3·5·7x^9

sin -1 x = x + 2 3 + ~ 5 + ~ 7 + 2. 4. 6. 8 9 + ... on [-1, 1 J.