Comment. Series quiz
Comment. This quiz is designed to test your knowledge of concepts in series such as convergence,
Comment. absolute convergence, and the various convergence tests.
Shuffle Questions.
Shuffle Answers.
Question. The series 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 ... is
Answer. Conditionally convergent but absolutely divergent.
Answer. Absolutely convergent but conditionally divergent.
Answer. Absolutely convergent and conditionally convergent.
Correct Answer. Absolutely divergent and conditionally divergent.
Answer. There is insufficient information to answer the question.
Question. The series 1 + 1/sqrt(2) + 1/sqrt(3) + 1/sqrt(4) + 1/sqrt(5) + 1/sqrt(6) ... is
Answer. Conditionally convergent but absolutely divergent.
Answer. Absolutely convergent but conditionally divergent.
Answer. Absolutely convergent and conditionally convergent.
Correct Answer. Absolutely divergent and conditionally divergent.
Answer. There is insufficient information to answer the question.
Question. The series 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 ... is
Correct Answer. Conditionally convergent but absolutely divergent.
Answer. Absolutely convergent but conditionally divergent.
Answer. Absolutely convergent and conditionally convergent.
Answer. Absolutely divergent and conditionally divergent.
Answer. There is insufficient information to answer the question.
Question. If x is a real number, the series x + x^2/2 + x^3/3 + x^4/4 + x^5/5 + x^6/6 ... is
Correct Answer. Absolutely convergent for x in (-1,1), conditionally convergent only for x = -1, and divergent elsewhere.
Answer. Absolutely convergent for x in (-1,1), conditionally convergent for x = -1 and +1, and divergent elsewhere.
Answer. Absolutely convergent for x in [-1,1] and divergent elsewhere.
Answer. Conditionally convergent only for x in (-1,1), and divergent elsewhere.
Answer. Conditionally convergent only for x in [-1,1], and divergent elsewhere.
Answer. Absolutely convergent for x in (-1,1], and divergent elsewhere.
Answer. Absolutely convergent for x in (-1,1], conditionally convergent for x = -1, and divergent elsewhere.
Question. If x is a real number, the series x + 2x^2 + 3x^3 + 4x^4 + 5x^5 + 6x^6 ... is
Answer. Absolutely convergent for x in (-1,1), conditionally convergent only for x = -1, and divergent elsewhere.
Answer. Absolutely convergent for x in (-1,1), conditionally convergent for x = -1 and +1, and divergent elsewhere.
Answer. Absolutely convergent for x in [-1,1] and divergent elsewhere.
Answer. Conditionally convergent only for x in (-1,1), and divergent elsewhere.
Answer. Conditionally convergent only for x in [-1,1], and divergent elsewhere.
Correct Answer. Absolutely convergent for x in (-1,1), and divergent elsewhere.
Answer. Absolutely convergent for x in (-1,1], conditionally convergent for x = -1, and divergent elsewhere.
Question. The series 1 - 1/sqrt(2) + 1/sqrt(3) - 1/sqrt(4) + 1/sqrt(5) - 1/sqrt(6) ... is
Correct Answer. Conditionally convergent but absolutely divergent.
Answer. Absolutely convergent but conditionally divergent.
Answer. Absolutely convergent and conditionally convergent.
Answer. Absolutely divergent and conditionally divergent.
Answer. There is insufficient information to answer the question.
Question. The series 1 - 1 + 1 - 1 + 1 - 1 ... is
Answer. Conditionally convergent but absolutely divergent.
Answer. Absolutely convergent but conditionally divergent.
Answer. Absolutely convergent and conditionally convergent.
Correct Answer. Absolutely divergent and conditionally divergent.
Answer. There is insufficient information to answer the question.
Question. The series 1.1 - 1.01 + 1.001 - 1.0001 + 1.00001 - 1.000001 ... is
Answer. Conditionally convergent but absolutely divergent.
Answer. Absolutely convergent but conditionally divergent.
Answer. Absolutely convergent and conditionally convergent.
Correct Answer. Absolutely divergent and conditionally divergent.
Answer. There is insufficient information to answer the question.
Question. The series 1/2 + 2/2^2 + 3/2^3 + 4/2^4 + ... + n/2^n + ... is
Answer. Conditionally convergent but absolutely divergent.
Answer. Absolutely convergent but conditionally divergent.
Correct Answer. Absolutely convergent and conditionally convergent.
Answer. Absolutely divergent and conditionally divergent.
Answer. There is insufficient information to answer the question.
Question. Let a_1 + a_2 + a_3 + ... be a series of real numbers. What does it mean for this series to be convergent?
Correct Answer. There is a real number L such that the sequence of partial sums a_1 + a_2 + ... + a_n converge to L.
Answer. There is a real number L such that the sequence of partial sums |a_1| + |a_2| + ... + |a_n| converge to L.
Answer. There is a real number L such that the sequence a_1, a_2, a_3, ... converges to L.
Answer. There is a real number L such that the sequence |a_1|, |a_2|, |a_3|, ... converges to L.
Answer. The sequence a_1, a_2, a_3, ... converges to 0.
Answer. We have a_{n+1} < a_n for every integer n >= 1.
Answer. We have |a_{n+1}| < |a_n| for every integer n >= 1.
Question. Let a_1 + a_2 + a_3 + ... be a series of real numbers. What does it mean for this series to be absolutely convergent?
Answer. There is a real number L such that the sequence of partial sums a_1 + a_2 + ... + a_n converge to L.
Correct Answer. There is a real number L such that the sequence of partial sums |a_1| + |a_2| + ... + |a_n| converge to L.
Answer. There is a real number L such that the sequence a_1, a_2, a_3, ... converges to L.
Answer. There is a real number L such that the sequence |a_1|, |a_2|, |a_3|, ... converges to L.
Answer. The sequence a_1, a_2, a_3, ... converges to 0.
Answer. The series is convergent, and all the terms a_1, a_2, ... in the series are non-negative.
Answer. We have |a_{n+1}| < |a_n| for every integer n >= 1.
Question. If a_1, a_2, ... converges to 0 as n tends to infinity, then we know the series a_1 + a_2 + ...
Answer. Must be conditionally convergent, but could be absolutely divergent.
Answer. Must be absolutely convergent and conditionally convergent.
Answer. Must be absolutely convergent and conditionally divergent.
Answer. Must be absolutely convergent, but could be conditionally divergent.
Answer. Must be absolutely divergent and conditionally divergent.
Answer. Must be absolutely divergent and conditionally convergent.
Correct Answer. No conclusion on either absolute or conditional convergence can be drawn.
Question. If a_1, a_2, ... does NOT converge to 0 as n tends to infinity, then we know the series a_1 + a_2 + ...
Answer. Must be conditionally convergent, but could be absolutely divergent.
Answer. Must be absolutely convergent and conditionally convergent.
Answer. Must be absolutely convergent and conditionally divergent.
Answer. Must be absolutely convergent, but could be conditionally divergent.
Correct Answer. Must be absolutely divergent and conditionally divergent.
Answer. Must be absolutely divergent and conditionally convergent.
Answer. No conclusion on either absolute or conditional convergence can be drawn.
Question. If a_1 + a_2 + ... is absolutely convergent, and |a_n| < b_n for all n, then b_1 + ... + b_n
Answer. Must be conditionally convergent, but could be absolutely divergent.
Answer. Must be absolutely convergent and conditionally convergent.
Answer. Must be absolutely convergent and conditionally divergent.
Answer. Must be absolutely divergent, but could be conditionally convergent.
Answer. Must be absolutely divergent and conditionally divergent.
Answer. Must be absolutely divergent and conditionally convergent.
Correct Answer. No conclusion on either absolute or conditional convergence can be drawn.
Question. If a_1 + a_2 + ... is absolutely convergent, and b_n < a_n for all n, then b_1 + ... + b_n
Comment. One needs to have |b_n| bounded by a_n before one can apply the Comparison test.
Answer. Must be conditionally convergent, but could be absolutely divergent.
Answer. Must be absolutely convergent and conditionally convergent.
Answer. Must be absolutely convergent and conditionally divergent.
Answer. Must be absolutely divergent, but could be conditionally convergent.
Answer. Must be absolutely divergent and conditionally divergent.
Answer. Must be absolutely divergent and conditionally convergent.
Correct Answer. No conclusion on either absolute or conditional convergence can be drawn.
Question. If a_1 + a_2 + ... is conditionally convergent, and |b_n| < a_n for all n, then b_1 + ... + b_n
Comment. Note that the a_n must be positive and so a_1 + a_2 + ... is not just conditionally convergent but also absolutely convergent.
Answer. Must be conditionally convergent, but could be absolutely divergent.
Correct Answer. Must be absolutely convergent and conditionally convergent.
Answer. Must be absolutely convergent and conditionally divergent.
Answer. Must be absolutely divergent, but could be conditionally convergent.
Answer. Must be absolutely divergent and conditionally divergent.
Answer. Must be absolutely divergent and conditionally convergent.
Answer. No conclusion on either absolute or conditional convergence can be drawn.
Question. If a_1 + a_2 + ... is absolutely divergent, and |a_n| < |b_n| for all n, then b_1 + ... + b_n
Answer. Must be conditionally convergent, but could be absolutely divergent.
Answer. Must be absolutely convergent and conditionally convergent.
Answer. Must be absolutely convergent and conditionally divergent.
Correct Answer. Must be absolutely divergent, but could be conditionally convergent.
Answer. Must be absolutely divergent and conditionally divergent.
Answer. Must be absolutely divergent and conditionally convergent.
Answer. No conclusion on either absolute or conditional convergence can be drawn.
Question. What are all the conditions required by the Alternating Series test to force a_1 + a_2 + ... to be conditionally convergent?
Correct Answer. The sequence a_n must be alternating in sign, |a_n| must be decreasing, and a_n must converge to 0.
Answer. The sequence a_n must be alternating in sign, and |a_n| must be decreasing.
Answer. The sequence a_2, a_4, a_6, ... must be decreasing, and the sequence a_1, a_3, a_5 must be increasing.
Answer. The sequence a_n must be alternating in sign, and |a_n| must converge to 0.
Answer. The sequence a_n must be alternating in sign, and a_n must converge.
Answer. The sequence a_n must be alternating in sign, and a_1 + a_2 + ... must be absolutely convergent.
Answer. The sequence a_n must be alternating in sign, a_n must be decreasing, and a_n must converge to 0.
Question. What condition is required by the Root test to force a_1 + a_2 + ... to be conditionally convergent?
Correct Answer. the lim sup of |a_n|^{1/n} must be strictly less than 1.
Comment. This will also make a_1 + a_2 + ... absolutely convergent as well.
Answer. the lim sup of a_n^{1/n} must be strictly less than 1.
Answer. the lim sup of |a_n|^{1/n} must be less than or equal to 1.
Answer. the lim inf of a_n^{1/n} must be less than or equal to 1.
Answer. the lim inf of |a_n|^{1/n} must be strictly less than 1.
Partially Correct Answer. the limit of |a_n|^{1/n} must exist and be strictly less than 1.
Comment. This will suffice to force convergence, but it is not the most general condition that the Root test can use.
Answer. |a_n|^{1/n} must be strictly less than 1 for each n.
Question. What condition is required by the Root test to force a_1 + a_2 + ... to be conditionally divergent?
Correct Answer. the lim sup of |a_n|^{1/n} must be strictly greater than 1.
Answer. the lim sup of a_n^{1/n} must be strictly greater than 1.
Answer. the lim inf of |a_n|^{1/n} must be greater than or equal to 1.
Partially Correct Answer. the lim inf of |a_n|^{1/n} must be strictly greater than 1.
Comment. This will suffice to force divergence, but it is not the most general condition that the Root test can use.
Partially Correct Answer. the limit of |a_n|^{1/n} must exist and be strictly greater than 1.
Comment. This will suffice to force divergence, but it is not the most general condition that the Root test can use.
Partially Correct Answer. |a_n|^{1/n} must be strictly greater than 1 for each n.
Comment. This will suffice to force divergence, but it is not the most general condition that the Root test can use.
Answer. The limit of the sequence |a_n|^{1/n} does not exist.
Question. Suppose the a_n are all non-zero. What condition is required by the Ratio test to force a_1 + a_2 + ... to be conditionally convergent?
Correct Answer. the lim sup of |a_{n+1}|/|a_n| must be strictly less than 1.
Comment. This will also make a_1 + a_2 + ... absolutely convergent as well.
Answer. the lim sup of a_{n+1}/a_n must be strictly less than 1.
Answer. the lim sup of |a_n|/|a_{n+1}| must be strictly less than to 1.
Answer. the lim inf of |a_{n+1}|/|a_n| must be less than or equal to 1.
Answer. the lim inf of |a_{n+1}|/|a_n| must be strictly less than 1.
Partially Correct Answer. the limit of |a_{n+1}|/|a_n| must exist and be strictly less than 1.
Comment. This will suffice to force convergence, but it is not the most general condition that the Root test can use.
Answer. |a_{n+1}|/|a_n| must be strictly less than 1 for each n.
Question. Suppose the a_n are all non-zero. What condition is required by the Ratio test to force a_1 + a_2 + ... to be conditionally divergent?
Comment. There are two distinct correct answers to this question (i.e. two different correct criteria to force divergence).
Correct Answer. the lim inf of |a_{n+1}|/|a_n| must be strictly greater than 1.
Answer. the lim inf of a_{n+1}/a_n must be strictly greater than 1.
Answer. the lim sup of |a_n|/|a_{n+1}| must be strictly greater than to 1.
Answer. the lim inf of |a_{n+1}|/|a_n| must be greater than or equal to 1.
Answer. the lim sup of |a_{n+1}|/|a_n| must be strictly greater than 1.
Partially Correct Answer. the limit of |a_{n+1}|/|a_n| must exist and be strictly greater than 1.
Comment. This will suffice to force convergence, but it is not the most general condition that the Root test can use.
Correct Answer. |a_{n+1}|/|a_n| must be greater than or equal to 1 for each n.
Question. Suppose that a_1 + a_2 + ... converges to L. If I rearrange the elements of this series arbitrarily, must the rearranged sum also converge to L?
Correct Answer. Yes if the sum is absolutely convergent, but not necessarily otherwise.
Answer. Yes if the sum is conditionally convergent, but not necessarily otherwise.
Answer. Yes in all cases.
Answer. No; only finite series can be rearranged arbitrarily.
Answer. The rearranged series will always converge, but may converge to something other than L.
Partially Correct Answer. Yes if all the terms in the series have the same sign, but not necessarily otherwise.
Comment. This is one way to ensure that rearrangements work correctly, but it is not the strongest claim one can make here.
Answer. Yes if the sequence a_1, a_2, a_3, ... converges to zero, but not necessarily otherwise.