1549901369-Elements_of_Real_Analysis__Denlinger_

8.6 Power Series 517

(b) Applying similar reasoning to Example 8.6.18 (b), we can show that
the Maclaurin series for tan-^1 x is a valid representation of tan-^1 x on the
interval [-1, l]. As a consequence we have the interesting result,

7r 1 1 1 1

4=l-3+5-7+9-···

(See Exercise 15.) From this we get a series representation for 7r, although a
very slowly converging one. D

EXERCISE SET 8.6

l. Prove Corollary 8.6.3.

2. Prove Theorem 8.6.7.


3. Prove Theorem 8.6.8.


4. Find the radius of convergence and the interval of convergence of each of
   the following power series.
   00 00 k'
   (a) Llnk(x + l)k (b) L
   2
   ~xk
   k=l k=O
   ~ xk (d) ~ (x + 5)k
   (c) ~ kk ~ k33k
   k=l k=l
   (e) f ~: (x - 3)k (f) f ( k;

1

) k (x + 4)k

k=O k=l

00 ( 3)k 00 1

(g) L k-+2(x+l)k (h) L1n2k(x-^2 )k

k~O ( k ) k k~l ( k ) k

(i) ~ cos

6

7r (x - 3)k (j) ~ 1 +sin : (x + 2)k

(k) f 1~2k (x + 2)k (1) f :! (x + 7)k

k=l k=l

(m) 1 - ~x + ~x^2 - ~x^3 + l ·^3.^5.^7 x^4 - · · · [See Exercise

2 2·4 2·4·6 2·4·6·8

8.3.13.]

x x2 x3 x4 x5 oo

(n) 1 + - + - + - + - + - + · · · = 2:::; akxk

2 32 23 34 25 k=O '

h {

( ~) k if k is odd }

w ere ak = k.

(~) if k is even

5. Prove Theorem 8.6.9.


6. Prove the claims made in Example 8.6.13.