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12. Given a sequence {a_n} which of the following statements provides a proper definition for convergence? a) The sequence {a_n} converges to L if there exists an ? > 0 and cutoff N ? ?, such that |a_n - L| < ? for all n ? N. b) The sequence {a_n} converges to L if there exists an ? > 0 such that for every cutoff N ? ?, we have |a_n - L| < ? for all n ? N. c) The sequence {a_n} converges to L if for each ? > 0 there is a cutoff N ? ?, such that |a_n - L| < ? for all n ? N. d) The sequence {a_n} converges to L if for each ? > 0 and for every cutoff N ? ?, we have |a_n - L| < ? for all n ? N. e) None of the above. 13. Note that |(4n+3)/(2n+1) - 2| = 1/(2n+1). Find a cutoff N ? ? so that if n ? N, then |(4n+3)/(2n+1) - 2| < 1/100. a) N = 48 b) N = 49 c) N = 50 d) N = 51 e) None of the above. 14. Assume that {a_n} is a sequence such that |a_{2k} - 1| < 1/2 and |a_{2k-1} + 1| < 1/2 for each k ? ?. Which of the following statements must be true? a) If {a_n} converges to L, then L would have to be in (1/2, 3/2). b) If {a_n} converges to L, then L would have to be in (-3/2, -1/2). c) The sequence {a_n} diverges. d) The sequence oscillates. e) None of the above.

          12. Given a sequence {a_n} which of the following statements provides a proper definition for convergence?
a) The sequence {a_n} converges to L if there exists an ? > 0 and cutoff N ? ?, such that |a_n - L| < ? for all n ? N.
b) The sequence {a_n} converges to L if there exists an ? > 0 such that for every cutoff N ? ?, we have |a_n - L| < ? for all n ? N.
c) The sequence {a_n} converges to L if for each ? > 0 there is a cutoff N ? ?, such that |a_n - L| < ? for all n ? N.
d) The sequence {a_n} converges to L if for each ? > 0 and for every cutoff N ? ?, we have |a_n - L| < ? for all n ? N.
e) None of the above.

13. Note that |(4n+3)/(2n+1) - 2| = 1/(2n+1). Find a cutoff N ? ? so that if n ? N, then |(4n+3)/(2n+1) - 2| < 1/100.
a) N = 48
b) N = 49
c) N = 50
d) N = 51
e) None of the above.

14. Assume that {a_n} is a sequence such that |a_{2k} - 1| < 1/2 and |a_{2k-1} + 1| < 1/2 for each k ? ?. Which of the following statements must be true?
a) If {a_n} converges to L, then L would have to be in (1/2, 3/2).
b) If {a_n} converges to L, then L would have to be in (-3/2, -1/2).
c) The sequence {a_n} diverges.
d) The sequence oscillates.
e) None of the above.

12. Given a sequence an which of the following statements provides a proper definition for convergence?
a) The sequence an converges to L if there exists an ? > 0 and cutoff N ? ?, such that |an - L| < ? for all n ? N.
b) The sequence an converges to L if there exists an ? > 0 such that for every cutoff N ? ?, we have |an - L| < ? for all n ? N.
c) The sequence an converges to L if for each ? > 0 there is a cutoff N ? ?, such that |an - L| < ? for all n ? N.
d) The sequence an converges to L if for each ? > 0 and for every cutoff N ? ?, we have |an - L| < ? for all n ? N.
e) None of the above.

13. Note that |(4n+3)/(2n+1) - 2| = 1/(2n+1). Find a cutoff N ? ? so that if n ? N, then |(4n+3)/(2n+1) - 2| < 1/100.
a) N = 48
b) N = 49
c) N = 50
d) N = 51
e) None of the above.

14. Assume that an is a sequence such that |a2k - 1| < 1/2 and |a2k-1 + 1| < 1/2 for each k ? ?. Which of the following statements must be true?
a) If an converges to L, then L would have to be in (1/2, 3/2).
b) If an converges to L, then L would have to be in (-3/2, -1/2).
c) The sequence an diverges.
d) The sequence oscillates.
e) None of the above.

Added by Kenneth T.

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