Unformatted text preview: 364 Section 9.1 69. continued
dx (b) k dt (a x)(b x) dx k dt (a x)(b x) 1 A (a x)(b x) a x Chapter 9 Infinite Series
kt
B b x C1 s Section 9.1 Power Series (pp. 457468)
Exploration 1
1. 1 2. x 3. 1 4. 1
1 3 1 A(b ( A x) B)x B(a bA x) aB Power Series for Other Functions
x3 x
4 x x
2 x2 x
3 2 ... ...
3 2 ( x)n ( 1) x ... (2x)
n ....
n n 1 .... .... ... ... Equating coefficients of like terms gives A B 0 and bA aB 1 2x (x 4x 8x (x 1) (x 1) ( 1)n(x 1)n .... 1)
1 n (x 3 1 (x 3 1)
1 (x 3 3 5. Solving the system simultaneously yields A
(a 1 a dx x)(b b 1 a b 1/(a b) dx a x ln a x a b 1 a 1 a b a x ln (a b)kt b x a x De(a b)kt b x b ln b x a b a b x x 1/(a b) dx b x 1 (x 3 1)2 .... 1)3 1)n ,B
x) This geometric series converges for which is equivalent to 0 gence is (0, 2). x 1 x 1 1, 2. The interval of conver C2 ln kt C C2 Exploration 2
1. 1 2. tan x
1 2 A Power Series for tan
x
6 1 x ln a b x x x 4
x ... dt t4
t5 5 x5 5 ( 1) x n 2n .... C2 C1 x 1 t2 0 1 x 0 (1
t3 3 x3 3 t2 t6
t7 7 x7 7 ... ... ... ( 1)nt 2n ( 1)n ...) dt
x 0 t 0. x Substitute t
a b a b 0, x D
x x a (a b)kt e b t 2n 1 ... 2n 1 2n 1 x .... ( 1)n 2n 1 3. The graphs of the first four partial sums appear to be converging on the interval ( 1, 1).
b)kt ab bx
b)kt abe (a b)
b)kt b)kt axe (a
b)kt b)kt x(ae (a x ab(e (a
1) b 1)
[ 5, 5] by [ 3, 3] ab(e (a ae (a 4. When x
e bkt Multiply the rational expression by bkt . e ab(e akt e bkt) x akt bkt ae be 1, the series becomes
1 5 1 7 1 1 3 ... ( 1)n 2n 1 .... This series does appear to converge. The terms are getting smaller, and because they alternate in sign they cause the partial sums to oscillate above and below a limit. The two calculator statements shown below will cause the successive partial sums to appear on the calculator each time the ENTER button is pushed. The partial sums will appear to be approaching a limit of /4 (which is tan 1(1)), although very slowly. ...
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN
 Infinite Series, Power Series

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