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.

 
solution

 

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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