1549901369-Elements_of_Real_Analysis__Denlinger_

526 Chapter 8 a Infinite Series of Real Numbers

J

dx
Proof. Recall that sin-^1 x = .Jl=X2' From Example 8.7. 11 , we have
1-x^2
the Maclaurin series

L

oo (-l)kl · 3 · 5 .... · (2k - 1)

= l+ (-l)k x2k.

2kk!

k=l

For lxl < 1, we integrate term-by-term and find that

J

dx - C + + ~ 1. 3. 5 ..... (2k - 1) 2k+l Since sin-1 0 0,

v'l - x2 - x ~ (2k + 1)2kk! x ·

k=l

C=O.

Therefore, when lxl < 1,

. -1 1 3 1. 3 5 1. 3. 5 7 1. 3. 5. 7
sm x =x +3·2·l! x +5.22.2!x +7·23·3!x + 9.24.4! x9+ .... (33)

We now test the endpoints of the interval (-1, 1) for convergence of this

series.

Test the endpoint x = 1. We apply Raabe's test (8.2.21):

R = lim k (1 -ak+I )

k-->oo ak

= lim k(l- 1 ·3·5 .. ·(2k-1)(2k+l). 2 ·4·6^00 ·(2k)(2k+l))

k-->oo 2 · 4 · 6 .. · (2k) (2k + 2) (2k + 3) 1 · 3 · 5 .. · (2k - 1)

= lim k (1 - (^2 k - l)

2

) = lim^6 k

2

+^5 k = ~. Since R > 1

k-->oo (2k + 2)(2k + 3) k-->oo 4k^2 + lOk + 6 2 '

Raabe's test tells us that the series converges.

Test the endpoint x = -1. Since the series (33) contains only odd-

degree terms, the partial sums for -x are the negatives of the partial sums for

x. Therefore, the series (33) converges for x = -1.

Therefore, the interval of convergence of the series (33) is [-1, l]. Since
sin-^1 x is continuous from the right at -1 and continuous from the left at 1,
Abel's theorem 8.6.19 guarantees that this power series converges to sin-^1 x
everywhere on [-1, l]. D

FURTHER THEORETICAL CONSIDERATIONS
Any function that is infinitely differentiable at c "has" a Taylor series in
the sense that all its Taylor coefficients exist. However, it would be wishful
thinking to conclude that such a function is necessarily analytic at c. For a