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

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. 
3.
A. 4 
B. 6 
C. 8 
D. 12 
E. infinity 
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. 3e^{x} + 2x  1 
B. e^{3x} + 2x + 1 
C. e^{3x}  x + 1 
D. 3e^{x}  x  1 
E. 3e^{x} + 5x + 5 
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. 
6. What is the Taylor series for f(x) = e^{x} about the point x = 1?
A. 
B. 
C. 
D. 
E. 
7. Let {a_{n}} be a sequence of positive real numbers such that a_{n+1}/a_{n} <= (n+4)/(2n+1) for all n. Then
A. 0 
B. 1/2 
C. 1 
D. 2 
E. 4 
8. Which of the following are subsequences of the sequence {a_{n}} 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. 
9. Let {a_{n}} be a sequence such that a_{0} = 1 and
(n^{2} + 2)a_{n+1}  (n^{2} + 1)pa_{n} = 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} 
10. Which of the following is an interval of convergence for the series
A. 
B. 
C. 
D. 
E. 
11. What is the Taylor series for the function f(x) = e^{2x+1} about x = 1?

B. 
C. 
D. 
E. 
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. 
13.
A. 388 
B. 392 
C. 440 
D. 1372 
E. 
14. Let a_{n} = nsin(3/n), for positive integers n. Then
A. 0 
B. 1 
C. 3 
D. 6 
E. 
15. Let a_{1} = 3/4 and a_{n+1} = (1/2)a_{n} for n = 1,2,. . What is
A. (1 + 2^{7})/2^{2} 
B. (2 + 2^{8}) 
C. ((1 + 2^{7})/2^{11}) 
D. (1 + 2^{7})/2^{8} 
E. (2^{6}  1)/2^{11} 
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. 
17. Let {a_{n}} be a geometric sequence for which a_{3} = 8 and a_{6} = 128. Then a_{1} =
A. 1/2 
B. 1 
C. 2^{1/3} 
D. 4^{1/3} 
E. 2 
18. What are all the values of x for which the infinite series
(x5) + 2(x5)^{2 }+ 3(x5)^{3 }+ 4(x5)^{4 }+ à

converges?
A. 
B. 
C. 
D. 
E. 
19. What is the interval of convergence of the power series
A. 
B. 
C. 
D. 
E. 
20. Let
be an alternating series for which each a_{n} > 0 and the limit of a_{n} 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. 
21.
A. e^{x}  1 
B. e^{x} 
C. e^{x}  1 
D. e^{x} 
E. xe^{x} 
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} 