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.

solution

 

3.

A. 4

B. 6

C. 8

D. 12

E. infinity

solution

 

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

solution

 

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.

solution

 

6. What is the Taylor series for f(x) = ex about the point x = 1?

A.

B.

C.

D.

E.

solution

 

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

solution

 

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.

solution

 

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}

solution

 

10. Which of the following is an interval of convergence for the series

A.

B.

C.

D.

E.

solution

 

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

 

B.

C.

D.

E.

solution

 

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.

solution

 

13.

A. 388

B. 392

C. 440

D. 1372

E.

solution

 

14. Let an = nsin(3/n), for positive integers n. Then

A. 0

B. 1

C. 3

D. 6

E.

solution

 

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

solution

 

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.

solution

 

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

solution

 

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 +

 

converges?

A.

B.

C.

D.

E.

solution

 

19. What is the interval of convergence of the power series

A.

B.

C.

D.

E.

solution

 

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.

solution

 

21.

A. e-x - 1

B. e-x

C. ex - 1

D. -e-x

E. -xe-x

solution

 

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}

solution