1.Let

for n = 1,2, . . .,.Which statement is true of the sequence {an}?

 A. It is bounded but does not converge. B. It converges to 0. C. It converges to a positive number. D. It diverges to infinity. E. It is unbounded and contains both arbitrarily large positive and arbitrarily large negative terms.
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2. For a real number alpha, consider the series

A necessary and sufficient condition for this series to be convergent is

 A. B. C. D. E.
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3.

 A. 4 B. 6 C. 8 D. 12 E. infinity
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4. Let f:R->R be a function with Taylor series converging to f(x) for all real numbers x. If f(0) = 2, f(0) = 2, and f(n)(0) = 3 for n >= 2, then f(x) =

 A. 3ex + 2x - 1 B. e3x + 2x + 1 C. e3x - x + 1 D. 3ex - x - 1 E. 3ex + 5x + 5
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5. Which of the following series converge?

 A. None B. I only C. II only D. III only E. The correct answer is not given by A, B, C, or D.
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6. What is the Taylor series for f(x) = ex about the point x = 1?

 A. B. C. D. E.
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7. Let {an} be a sequence of positive real numbers such that an+1/an <= (n+4)/(2n+1) for all n. Then

 A. 0 B. 1/2 C. 1 D. 2 E. 4
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8. Which of the following are subsequences of the sequence {an} defined by

 A. None B. I only C. II only D. III only E. The correct answer is not given by A, B, C, or D.
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9. Let {an} be a sequence such that a0 = 1 and

 (n2 + 2)an+1 - (n2 + 1)pan = 0

for n >= 0. What are all the values of p for which the series

is absolutely convergent?

 A. {p | p > 1} B. {p | p < -1} C. {p | |p| < 1} D. {p | |p| < 2} E. {p | |p| < 1/2}
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10. Which of the following is an interval of convergence for the series

 A. B. C. D. E.
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11. What is the Taylor series for the function f(x) = e2x+1 about x = -1?

 B. C. D. E.
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12. Which of the following are sufficient conditions for the convergence of

 A. None B. I only C. II only D. III only E. The correct answer is not given by A, B, C, or D.
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13.

 A. 388 B. 392 C. 440 D. 1372 E.
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14. Let an = nsin(3/n), for positive integers n. Then

 A. 0 B. 1 C. 3 D. 6 E.
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15. Let a1 = 3/4 and an+1 = (-1/2)an for n = 1,2,. . What is

 A. (1 + 27)/22 B. -(2 + 28) C. -((1 + 27)/211) D. (1 + 27)/28 E. (26 - 1)/211
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16. Which of the following series converge?

 A. I and II only B. I and III only C. II and III only D. I, II, and III E. The correct answer is not given by A, B, C, or D.
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17. Let {an} be a geometric sequence for which a3 = 8 and a6 = 128. Then a1 =

 A. 1/2 B. 1 C. 21/3 D. 41/3 E. 2
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18. What are all the values of x for which the infinite series

 (x-5) + 2(x-5)2 + 3(x-5)3 + 4(x-5)4 + à

converges?

 A. B. C. D. E.
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19. What is the interval of convergence of the power series

 A. B. C. D. E.
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20. Let

be an alternating series for which each an > 0 and the limit of an as n goes to infinity is equal to zero. Which of the following conditions is sufficient to guarantee that S converges?

 A. B. C. D. E.
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21.

 A. e-x - 1 B. e-x C. ex - 1 D. -e-x E. -xe-x
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22. What is the set of limit points of the sequence

 A. {0} B. {1} C. {-1,1} D. {0,1} E. {0,1,-1}

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