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Quiz: Sequences

Page history last edited by RH 11 years, 7 months ago

This quiz is designed to test your knowledge of concepts in sequences such as convergence, Cauchy convergence, boundedness, limsup, liminf, limits, limit points, etc.

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

  1. In the rationals Formula, the sequence 1, 1/2, 1/3, 1/4, 1/5, 1/6, ... is
    1. Bounded.
    2. Convergent.
    3. A Cauchy sequence.
    4. A and C.
    5. B and C.
    6. A, B, and C.
      • CORRECT.
    7. None of the above.
  2. In the rationals Formula, the sequence 1, -1, 1, -1, 1, -1, ... is
    1. Bounded.
      • CORRECT.
    2. Convergent.
    3. A Cauchy sequence.
    4. A and C.
    5. B and C.
    6. A, B, and C.
    7. None of the above.
  3. In the reals Formula, the sequence 1, -1/2, 1/3, -1/4, 1/5, -1/6, ... is
    1. Bounded.
    2. Convergent.
    3. A Cauchy sequence.
    4. A and C.
    5. B and C.
    6. A, B, and C.
      • CORRECT.
    7. None of the above.
  4. In the rationals Formula, the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... is
    1. Bounded.
    2. Convergent.
    3. A Cauchy sequence.
    4. A and C.
      • CORRECT.
    5. B and C.
    6. A, B, and C.
    7. None of the above.
  5. In the reals Formula, the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... is
    1. Bounded.
    2. Convergent.
    3. A Cauchy sequence.
    4. A and C.
    5. B and C.
    6. A, B, and C.
      • CORRECT.
    7. None of the above.
  6. In the rationals Formula, the sequence 1, -2, 3, -4, 5, -6, ...is
    1. Bounded.
    2. Convergent.
    3. A Cauchy sequence.
    4. A and C.
    5. B and C.
    6. A, B, and C.
    7. None of the above.
      • CORRECT.
  7. In the rationals Formula, the sequence 0.9, -0.99, 0.999, -0.9999, 0.99999, -0.999999, ... is
    1. Bounded.
      • CORRECT.
    2. Convergent.
    3. A Cauchy sequence.
    4. A and C.
    5. B and C.
    6. A, B, and C.
    7. None of the above.
  8. The lim sup of the sequence 1.1, -1.01, 1.001, -1.0001, 1.00001, ... is
    1. 1
      • CORRECT.
    2. 1.1
    3. +1 and -1
    4. -1
    5. Does not exist
    6. Formula
  9. The lim inf of the sequence 0, -1, 0, -2, 0, -3, 0, -4, 0, -5, ... is
    1. Formula
      • CORRECT.
    2. Formula
    3. 0
    4. -1
    5. -5
    6. Does not exist
  10. The sup of the sequence 1, -1/2, 1/3, -1/4, 1/5, -1/6, ... is
    1. 1
      • CORRECT.
    2. 0
    3. -1/2
    4. Formula
    5. Does not exist
    6. Formula
  11. The limit points of the sequence 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ... are
    1. Just 0
      • CORRECT.
    2. 0 and Formula
    3. Formula
    4. No limit points exist.
    5. All the points 0, 1, 2, 3, ... are limit points.
    6. All positive numbers are limit points.
  12. The limit points of the sequence Formula for Formula are
    1. Just 1/2
    2. 0, -1/3, 2/5, -3/7, ...
    3. 0 and 1/2
    4. -1/2 and 1/2
      • CORRECT.
    5. 1 and -1
    6. 1/2 and 1
    7. No limit points exist.
  13. The inf of the sequence Formula for Formula is
    1. 0
      • CORRECT.
    2. 1
    3. 3/2
    4. 0 and 1
    5. Does not exist
    6. Formula
  14. The least upper bound of the set Formula is
    1. Formula
      • CORRECT.
    2. 0
    3. 2
    4. 4
    5. Formula
    6. Formula
    7. Does not exist
  15. The supremum of the set Formula is
    1. Formula
    2. 0
    3. 2
    4. 4
    5. Formula
      • CORRECT.
    6. Formula
    7. Does not exist
      • PARTIALLY. While it is true that the supremum does not exist as a real number, it does exist in the extended real number system (which includes Formula and Formula).
  16. The infimum of the set Formula is
    1. Formula
    2. Formula
      • CORRECT.
    3. 0
    4. 4
    5. Formula
    6. Formula
    7. Does not exist
  17. The least upper bound of the set Formula is
    1. Formula
    2. 0
    3. 2
    4. 4
    5. Formula
      • CORRECT.
    6. Formula
    7. Does not exist
      • PARTIALLY. While it is true that the least upper bound (or supremum) does not exist as a real number, it does exist in the extended real number system (which includes Formula and Formula).
  18. The infimum of the set Formula is
    1. Formula
    2. 0
    3. 2
    4. 4
      • CORRECT.
    5. Formula
    6. Formula
    7. Does not exist
  19. The infimum of the empty set is
    1. 0
    2. An arbitrary real number
    3. The empty set
    4. Positive infinity
      • CORRECT. It may seem unintuitive, but every real number is a lower bound for the empty set, and thus the greatest lower bound (infimum) is Formula.
    5. Formula
    6. Does not exist
  20. The supremum of the empty set is
    1. 0
    2. An arbitrary real number
    3. The empty set
    4. Negative infinity
      • CORRECT. It may seem unintuitive, but every real number is an upper bound for the empty set, and thus the least upper bound (supremum) is Formula.
    5. Formula
    6. Does not exist
  21. Let Formula be a sequence of real numbers. What does it mean for Formula to converge to a limit Formula as Formula?
    1. For every Formula, there exists an Formula such that Formula for all Formula.
      • CORRECT.
    2. The sequence Formula is decreasing, i.e. Formula for all Formula.
    3. For every Formula, there exists an Formula such that Formula for all Formula.
    4. For every Formula, and every Formula, we have Formula for all Formula.
    5. There exists Formula such that for every Formula, we have Formula for all Formula.
    6. There exists Formula and there exists Formula such that Formula for all Formula.
    7. For every Formula, there exists an Formula such that Formula for infinitely many Formula.
      • INCORRECT. This is what it means for Formula to be a limit point of Formula.
  22. Let Formula be a sequence of real numbers. What does it mean for the sequence Formula to have Formula as a limit point?
    1. For every Formula, there exists an Formula such that Formula for all Formula.
      • INCORRECT. This is what it means for Formula to converge to Formula, which is a slightly different concept.
    2. For every Formula, there exists an Formula and Formula such that Formula.
    3. For every Formula, and every Formula, there exists Formula such that Formula.
    4. There exists Formula such that for every Formula, there exists Formula such that Formula.
    5. There exists Formula and there exists Formula such that Formula for all Formula.
    6. For every Formula and Formula, there exists Formula such that Formula.
      • CORRECT.
  23. Let Formula be a sequence of real numbers. What does it mean for the sequence Formula to be bounded?
    1. There exists an Formula such that Formula for all Formula.
      • CORRECT.
    2. There exists an Formula such that Formula for some Formula.
    3. For each Formula there exists an Formula such that Formula.
    4. For every Formula and every Formula we have Formula.
    5. There exists an Formula such that Formula or Formula for each Formula.
    6. There exists an Formula and there exists Formula such that Formula.
    7. The sequence Formula either has a lower bound Formula, or an upper bound Formula, or both.
      • INCORRECT. One needs _both_ an upper bound and a lower bound in order to be bounded.
  24. Let Formula be a sequence of real numbers. What does it mean for Formula to be a Cauchy sequence?
    1. For every Formula, there exists an Formula such that Formula for all Formula.
      • CORRECT.
    2. The sequence Formula is decreasing, i.e. Formula for all Formula.
    3. For every Formula, there exists an Formula such that Formula for all Formula.
    4. For every Formula, and every Formula, we have Formulan,m \ge N$.
    5. There exists Formula such that for every Formula, we have Formula for all Formula.
    6. There exists Formula and there exists Formula such that Formula for all Formula.
    7. For every Formula, there exists an Formula such that Formula for infinitely many Formula.
  25. Let Formula be a subset of the reals Formula. What does it mean when we say that Formula is the supremum of Formula?
    1. Formula is an upper bound for Formula, and for any other upper bound Formula of Formula we have Formula.
      • CORRECT.
    2. Formula is the element in Formula which is larger than all the others.
    3. Formula is the lim sup of every sequence in Formula.
    4. For every Formula, we have Formula.
    5. For every Formula, we have Formula.
      • INCORRECT. This says that x is an upper bound for Formula, but does not say that it is the least upper bound for Formula.
    6. Formula is an upper bound for Formula, and is larger than all other upper bounds for Formula.
    7. Every number less than Formula lies in Formula, and every number greater than Formula lies outside of Formula.
  26. Let Formula be a subset of the reals Formula. What does it mean when we say that Formula is the infimum of Formula?
    1. Formula is a lower bound for Formula, and is larger than all other lower bounds for Formula.
      • CORRECT.
    2. Formula is a lower bound for Formula, and is smaller than all other lower bounds for Formula.
    3. Formula is the element in Formula which is smaller than all the others.
    4. Formula is the lim inf of every sequence in Formula.
    5. For every Formula, we have Formula.
    6. For every Formula, we have Formula.
      • INCORRECT. This says that Formula is an lower bound for Formula, but does not say that it is the greatest lower bound for Formula.
    7. Every number greater than Formula lies in Formula, and every number less than Formula lies outside of Formula.

 

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