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Unformatted text preview: 384 Section 9.4
an an (n 1)! x 4 n n! x 4 n
1 30. lim
n 1 lim
n 35. This is a geometric series with first term a common ratio r
(x 4 1)
2 1 and (x 4 1)2 . It converges only when lim (n
n 1) x 4) 4 1, so the interval of convergence is 3.
a 1 r 1 4 x2 x2 1
(x 4 1)2 (x 1 4, so the radius of Sum x The series converges only for x convergence is 0.
an an ( 2)
n 1 31. lim
n 1 lim
n (n 2) x 1 2n (n 1) x 1 n n 1 4 (x 1)2 lim 2 x
n 1 4 2x 4 2x 3 2x 1 1 3 The series converges for x x 1
an an 1 and diverges for 2 1 1 , so the radius of convergence is . 2 2 36. This is a geometric series with first term a common ratio r
(x 9 1)2 (x 9 1)2 1 and . It converges only when 1, so the interval of convergence is 2.
a 1 r 1 32. lim
n 1 lim
n 4x (n 5 1)3/2 2n 3 n 4x 3/2 5 2n 1 4 Sum x lim (4x
n 5)2 (4x 5)2 5)
2 The series converges for (4x to 4x x
5 4 an an
1 1, which is equivalent 1 9 (x 9 1)2 5 1 and diverges for 4 1 1 . The radius of convergence is . 4 4 1, or x 5 4 9 (x 1)2 9 2x x2 9 2x 8 x2 8 33. lim
n lim
n x n n 1 n x
n 37. This is a geometric series with first term a common ratio r
x 2 x 2 1 and 1 lim x
n 1. It converges only when x 0 The series converges for x x
an an 1 x 16.
a 1 1, so the interval of convergence is 1 and diverges for Sum
r 1 1
x 2 2 1 4 x 1, so the radius of convergence is 1.
x 2n 2 2n
1 3 34. lim
n 1 lim
n 2n x 2 2n
1 38. This is a geometric series with first term a common ratio r 1 and 1, 1 (x 2)2 n 2 1 (x 2)2 2 1 The series converges for (x 2 ln x. It converges only when ln x
1 e lim so the interval of convergence is Sum 2)2 1, which is
a 1 r 1 1 ln x x e. 39. This is a geometric series with first term a common ratio
x2 3 1 1 and
x2 3 1 equivalent to x x 2 2 2, and diverges for 2. . It converges only when 2
3 (x 2 1, 2. The radius of convergence is so the interval of convergence is Sum
a 1 r 1 1
x2 3 1 x 2.
3 3 1) 4 x2 ...
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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
 Geometric Series

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