Chapter 10

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January 31, 2005 12:53

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CHAPTER 10

Infinite Series EXERCISE SET 10.1 1. (a)

1

(b)

3n−1

(−1)n−1 3n−1

(c)

2. (a) (−r)n−1 ; (−r)n

2n − 1 2n

(d)

n2 π 1/(n+1)

(b) (−1)n+1 rn ; (−1)n rn+1 (b) 1, −1, 1, −1

3. (a) 2, 0, 2, 0

(c) 2(1 + (−1)n ); 2 + 2 cos nπ (b) (2n − 1)!

4. (a) (2n)! 5. 1/3, 2/4, 3/5, 4/6, 5/7, . . . ; lim

n→+∞

n = 1, converges n+2

n2 = +∞, diverges n→+∞ 2n + 1

6. 1/3, 4/5, 9/7, 16/9, 25/11, . . . ; lim

7. 2, 2, 2, 2, 2, . . .; lim 2 = 2, converges n→+∞

1 1 1 1 8. ln 1, ln , ln , ln , ln , . . . ; lim ln(1/n) = −∞, diverges n→+∞ 2 3 4 5 9.

ln 1 ln 2 ln 3 ln 4 ln 5 , , , , ,...; 1 2 3 4 5 ln x ln n 1 = lim = 0 apply L’Hˆ opital’s Rule to , converges lim n→+∞ n n→+∞ n x

10. sin π, 2 sin(π/2), 3 sin(π/3), 4 sin(π/4), 5 sin(π/5), . . . ; sin(π/n) (−π/n2 ) cos(π/n) = lim = π, converges n→+∞ n→+∞ 1/n −1/n2

lim n sin(π/n) = lim

n→+∞

11. 0, 2, 0, 2, 0, . . . ; diverges (−1)n+1 = 0, converges n→+∞ n2

12. 1, −1/4, 1/9, −1/16, 1/25, . . . ; lim

13. −1, 16/9, −54/28, 128/65, −250/126, . . . ; diverges because odd-numbered terms approach −2, even-numbered terms approach 2. 14. 1/2, 2/4, 3/8, 4/16, 5/32, . . . ; lim

n→+∞

n 1 = 0, converges = lim n n→+∞ 2 ln 2 2n

15. 6/2, 12/8, 20/18, 30/32, 42/50, . . . ; lim

n→+∞

1 (1 + 1/n)(1 + 2/n) = 1/2, converges 2

16. π/4, π 2 /42 , π 3 /43 , π 4 /44 , π 5 /45 , . . . ; lim (π/4)n = 0, converges n→+∞

17. cos(3), cos(3/2), cos(1), cos(3/4), cos(3/5), . . . ; lim cos(3/n) = 1, converges n→+∞

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