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Quiz: Series

Page history last edited by Terence Tao 9 years, 10 months ago

This quiz is designed to test your knowledge of concepts in series such as convergence, absolute convergence, and the various convergence tests.

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

  1. The series 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 ... is
    1. Conditionally convergent but absolutely divergent.
    2. Absolutely convergent but conditionally divergent.
    3. Absolutely convergent and conditionally convergent.
    4. Absolutely divergent and conditionally divergent.
      • CORRECT.
    5. There is insufficient information to answer the question.
  2. The series Formula is
    1. Conditionally convergent but absolutely divergent.
    2. Absolutely convergent but conditionally divergent.
    3. Absolutely convergent and conditionally convergent.
    4. Absolutely divergent and conditionally divergent.
      • CORRECT.
    5. There is insufficient information to answer the question.
  3. The series 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 ... is
    1. Conditionally convergent but absolutely divergent.
      • CORRECT.
    2. Absolutely convergent but conditionally divergent.
    3. Absolutely convergent and conditionally convergent.
    4. Absolutely divergent and conditionally divergent.
    5. There is insufficient information to answer the question.
  4. If Formula is a real number, the series Formula is
    1. Absolutely convergent for Formula, conditionally convergent only for Formula, and divergent elsewhere.
      • CORRECT.
    2. Absolutely convergent for Formula, conditionally convergent for Formula and Formula, and divergent elsewhere.
    3. Absolutely convergent for Formula and divergent elsewhere.
    4. Conditionally convergent only for Formula, and divergent elsewhere.
    5. Conditionally convergent only for Formula, and divergent elsewhere.
    6. Absolutely convergent for Formula, and divergent elsewhere.
    7. Absolutely convergent for Formula, conditionally convergent for Formula, and divergent elsewhere.
  5. If Formula is a real number, the series Formula is
    1. Absolutely convergent for Formula, conditionally convergent only for Formula, and divergent elsewhere.
    2. Absolutely convergent for Formula, conditionally convergent for Formula and Formula, and divergent elsewhere.
    3. Absolutely convergent for Formula and divergent elsewhere.
    4. Conditionally convergent only for Formula, and divergent elsewhere.
    5. Conditionally convergent only for Formula, and divergent elsewhere.
    6. Absolutely convergent for Formula, and divergent elsewhere.
      • CORRECT.
    7. Absolutely convergent for Formula, conditionally convergent for Formula, and divergent elsewhere.
  6. The series Formula is
    1. Conditionally convergent but absolutely divergent.
      • CORRECT.
    2. Absolutely convergent but conditionally divergent.
    3. Absolutely convergent and conditionally convergent.
    4. Absolutely divergent and conditionally divergent.
    5. There is insufficient information to answer the question.
  7. The series 1 - 1 + 1 - 1 + 1 - 1 ... is
    1. Conditionally convergent but absolutely divergent.
    2. Absolutely convergent but conditionally divergent.
    3. Absolutely convergent and conditionally convergent.
    4. Absolutely divergent and conditionally divergent.
      • CORRECT.
    5. There is insufficient information to answer the question.
  8. The series 1.1 - 1.01 + 1.001 - 1.0001 + 1.00001 - 1.000001 ... is
    1. Conditionally convergent but absolutely divergent.
    2. Absolutely convergent but conditionally divergent.
    3. Absolutely convergent and conditionally convergent.
    4. Absolutely divergent and conditionally divergent.
      • CORRECT.
    5. There is insufficient information to answer the question.
  9. The series Formula is
    1. Conditionally convergent but absolutely divergent.
    2. Absolutely convergent but conditionally divergent.
    3. Absolutely convergent and conditionally convergent.
      • CORRECT.
    4. Absolutely divergent and conditionally divergent.
    5. There is insufficient information to answer the question.
  10. Let Formula be a series of real numbers. What does it mean for this series to be convergent?
    1. There is a real number Formula such that the sequence of partial sums Formula converge to Formula.
      • CORRECT.
    2. There is a real number Formula such that the sequence of partial sums Formula converge to Formula.
    3. There is a real number Formula such that the sequence Formula converges to Formula.
    4. There is a real number Formula such that the sequence Formula converges to Formula.
    5. The sequence Formula converges to 0.
    6. We have Formula for every integer Formula.
    7. We have Formula for every integer Formula.
  11. Let Formula be a series of real numbers. What does it mean for this series to be absolutely convergent?
    1. There is a real number Formula such that the sequence of partial sums Formula converge to Formula.
    2. There is a real number Formula such that the sequence of partial sums Formula converge to Formula.
      • CORRECT.
    3. There is a real number Formula such that the sequence Formula converges to Formula.
    4. There is a real number Formula such that the sequence Formula converges to Formula.
    5. The sequence Formula converges to 0.
    6. The series is convergent, and all the terms Formula in the series are non-negative.
    7. We have Formula for every integer Formula.
  12. If Formula converges to 0 as Formula tends to infinity, then we know the series Formula
    1. Must be conditionally convergent, but could be absolutely divergent.
    2. Must be absolutely convergent and conditionally convergent.
    3. Must be absolutely convergent and conditionally divergent.
    4. Must be absolutely convergent, but could be conditionally divergent.
    5. Must be absolutely divergent and conditionally divergent.
    6. Must be absolutely divergent and conditionally convergent.
    7. No conclusion on either absolute or conditional convergence can be drawn.
      • CORRECT.
  13. If Formula does NOT converge to 0 as n tends to infinity, then we know the series Formula
    1. Must be conditionally convergent, but could be absolutely divergent.
    2. Must be absolutely convergent and conditionally convergent.
    3. Must be absolutely convergent and conditionally divergent.
    4. Must be absolutely convergent, but could be conditionally divergent.
    5. Must be absolutely divergent and conditionally divergent.
      • CORRECT.
    6. Must be absolutely divergent and conditionally convergent.
    7. No conclusion on either absolute or conditional convergence can be drawn.
  14. If Formula is absolutely convergent, and Formula for all Formula, then Formula
    1. Must be conditionally convergent, but could be absolutely divergent.
    2. Must be absolutely convergent and conditionally convergent.
    3. Must be absolutely convergent and conditionally divergent.
    4. Must be absolutely divergent, but could be conditionally convergent.
    5. Must be absolutely divergent and conditionally divergent.
    6. Must be absolutely divergent and conditionally convergent.
    7. No conclusion on either absolute or conditional convergence can be drawn.
      • CORRECT.
  15. If Formula is absolutely convergent, and Formula for all Formula, then Formula 
    1. Must be conditionally convergent, but could be absolutely divergent.
    2. Must be absolutely convergent and conditionally convergent.
      • One needs to have Formula bounded by Formula before one can apply the Comparison test. 
    3. Must be absolutely convergent and conditionally divergent.
    4. Must be absolutely divergent, but could be conditionally convergent.
    5. Must be absolutely divergent and conditionally divergent.
    6. Must be absolutely divergent and conditionally convergent.
    7. No conclusion on either absolute or conditional convergence can be drawn.
      • CORRECT.
  16. If Formula is conditionally convergent, and Formula for all Formula, then Formula 
    1. Must be conditionally convergent, but could be absolutely divergent.
    2. Must be absolutely convergent and conditionally convergent.
      • CORRECT. Note that the Formula must be positive and so Formula is not just conditionally convergent but also absolutely convergent.
    3. Must be absolutely convergent and conditionally divergent.
    4. Must be absolutely divergent, but could be conditionally convergent.
    5. Must be absolutely divergent and conditionally divergent.
    6. Must be absolutely divergent and conditionally convergent.
    7. No conclusion on either absolute or conditional convergence can be drawn.
  17. If Formula is absolutely divergent, and Formula for all Formula, then Formula
    1. Must be conditionally convergent, but could be absolutely divergent.
    2. Must be absolutely convergent and conditionally convergent.
    3. Must be absolutely convergent and conditionally divergent.
    4. Must be absolutely divergent, but could be conditionally convergent.
      • CORRECT.
    5. Must be absolutely divergent and conditionally divergent.
    6. Must be absolutely divergent and conditionally convergent.
    7. No conclusion on either absolute or conditional convergence can be drawn.
  18. What are all the conditions required by the Alternating Series test to force Formula to be conditionally convergent?
    1. The sequence Formula must be alternating in sign, Formula must be decreasing, and Formula must converge to 0.
      • CORRECT.
    2. The sequence Formula must be alternating in sign, and Formulamust be decreasing.
    3. The sequence Formula must be decreasing, and the sequence Formula must be increasing.
    4. The sequence Formula must be alternating in sign, and Formula must converge to 0.
    5. The sequence Formula must be alternating in sign, and Formula must converge.
    6. The sequence Formula must be alternating in sign, and Formula must be absolutely convergent.
    7. The sequence Formula must be alternating in sign, Formula must be decreasing, and Formula must converge to 0.
  19. What condition is required by the Root test to force Formula to be conditionally convergent?
    1. the lim sup of Formula must be strictly less than 1.
      • CORRECT. This will also make Formula absolutely convergent as well.
    2. the lim sup of Formula must be strictly less than 1.
    3. the lim sup of Formula must be less than or equal to 1.
    4. the lim inf of Formula must be less than or equal to 1.
    5. the lim inf of Formula must be strictly less than 1.
    6. the limit of Formula must exist and be strictly less than 1.
      • PARTIALLY. This will suffice to force convergence, but it is not the most general condition that the Root test can use.
    7. Formula must be strictly less than 1 for each n.
  20. What condition is required by the Root test to force Formula to be conditionally divergent?
    1. the lim sup of Formula must be strictly greater than 1.
      • CORRECT.
    2. the lim sup of Formula must be strictly greater than 1.
    3. the lim inf of Formula must be greater than or equal to 1.
    4. the lim inf of Formula must be strictly greater than 1.
      • PARTIALLY. This will suffice to force divergence, but it is not the most general condition that the Root test can use.
    5. the limit of Formula must exist and be strictly greater than 1.
      • PARTIALLY. This will suffice to force divergence, but it is not the most general condition that the Root test can use.
    6. Formula must be strictly greater than 1 for each Formula.
      • PARTIALLY. This will suffice to force divergence, but it is not the most general condition that the Root test can use.
    7. The limit of the sequence Formula does not exist.
  21. Suppose the a_n are all non-zero. What condition is required by the Ratio test to force Formula to be conditionally convergent?
    1. the lim sup of Formula must be strictly less than 1.
      • CORRECT. This will also make Formula absolutely convergent as well.
    2. the lim sup of Formula must be strictly less than 1.
    3. the lim sup of Formula must be strictly less than to 1.
    4. the lim inf of Formula must be less than or equal to 1.
    5. the lim inf of Formula must be strictly less than 1.
    6. the limit of Formula must exist and be strictly less than 1.
      • PARTIALLY. This will suffice to force convergence, but it is not the most general condition that the Root test can use.
    7. Formula must be strictly less than 1 for each Formula.
  22. Suppose the Formula are all non-zero. What condition is required by the Ratio test to force Formula to be conditionally divergent? (There are two distinct correct answers to this question (i.e. two different correct criteria to force divergence).)
    1. the lim inf of Formula must be strictly greater than 1.
      • CORRECT.
    2. the lim inf of Formula must be strictly greater than 1.
    3. the lim sup of Formula must be strictly greater than to 1.
    4. the lim inf of Formula must be greater than or equal to 1.
    5. the lim sup of Formula must be strictly greater than 1.
    6. the limit of Formula must exist and be strictly greater than 1.
      • PARTIALLY. This will suffice to force convergence, but it is not the most general condition that the Root test can use.
    7. Formula must be greater than or equal to 1 for each n.
      • CORRECT.
  23. Suppose that Formula converges to Formula. If I rearrange the elements of this series arbitrarily, must the rearranged sum also converge to Formula?
    1. Yes if the sum is absolutely convergent, but not necessarily otherwise.
      • CORRECT.
    2. Yes if the sum is conditionally convergent, but not necessarily otherwise.
    3. Yes in all cases.
    4. No; only finite series can be rearranged arbitrarily.
    5. The rearranged series will always converge, but may converge to something other than L.
    6. Yes if all the terms in the series have the same sign, but not necessarily otherwise.
      • PARTIALLY. This is one way to ensure that rearrangements work correctly, but it is not the strongest claim one can make here.
    7. Yes if the sequence Formula converges to zero, but not necessarily otherwise.

 

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