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

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