Comment. Sequences quiz
Comment. This quiz is designed to test your knowledge of concepts in sequences such as convergence,
Comment. Cauchy convergence, boundedness, limsup, liminf, limits, limit points, etc.
Shuffle Questions.
Don't Shuffle Answers.
Question. In the rationals Q, the sequence 1, 1/2, 1/3, 1/4, 1/5, 1/6, ... is
Answer. Bounded.
Answer. Convergent.
Answer. A Cauchy sequence.
Answer. A and C.
Answer. B and C.
Correct Answer. A, B, and C.
Answer. None of the above.
Question. In the rationals Q, the sequence 1, -1, 1, -1, 1, -1, ... is
Correct Answer. Bounded.
Answer. Convergent.
Answer. A Cauchy sequence.
Answer. A and C.
Answer. B and C.
Answer. A, B, and C.
Answer. None of the above.
Question. In the reals R, the sequence 1, -1/2, 1/3, -1/4, 1/5, -1/6, ... is
Answer. Bounded.
Answer. Convergent.
Answer. A Cauchy sequence.
Answer. A and C.
Answer. B and C.
Correct Answer. A, B, and C.
Answer. None of the above.
Question. In the rationals Q, the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159 ... is
Answer. Bounded.
Answer. Convergent.
Answer. A Cauchy sequence.
Correct Answer. A and C.
Answer. B and C.
Answer. A, B, and C.
Answer. None of the above.
Question. In the reals R, the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159 ... is
Answer. Bounded.
Answer. Convergent.
Answer. A Cauchy sequence.
Answer. A and C.
Answer. B and C.
Correct Answer. A, B, and C.
Answer. None of the above.
Question. In the rationals Q, the sequence 1, -2, 3, -4, 5, -6, ... is
Answer. Bounded.
Answer. Convergent.
Answer. A Cauchy sequence.
Answer. A and C.
Answer. B and C.
Answer. A, B, and C.
Correct Answer. None of the above.
Question. In the rationals Q, the sequence 0.9, -0.99, 0.999, -0.9999, 0.99999, -0.999999, ... is
Correct Answer. Bounded.
Answer. Convergent.
Answer. A Cauchy sequence.
Answer. A and C.
Answer. B and C.
Answer. A, B, and C.
Answer. None of the above.
// Now we begin the questions where it is safe to shuffle answers
Question. The lim sup of the sequence 1.1, -1.01, 1.001, -1.0001, 1.00001, ... is
Shuffle Answers.
Correct Answer. 1
Answer. 1.1
Answer. +1 and -1
Answer. -1
Answer. Does not exist
Answer. Positive infinity
Question. The lim inf of the sequence 0, -1, 0, -2, 0, -3, 0, -4, 0, -5, ... is
Correct Answer. Negative infinity
Answer. Positive infinity
Answer. 0
Answer. -1
Answer. -5
Answer. Does not exist
Question. The sup of the sequence 1, -1/2, 1/3, -1/4, 1/5, -1/6, ... is
Correct Answer. 1
Answer. 0
Answer. -1/2
Answer. ln(2)
Answer. Does not exist
Answer. Positive infinity
Question. The limit points of the sequence 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ... are
Correct Answer. Just 0
Answer. 0 and infinity
Answer. infinity
Answer. No limit points exist.
Answer. All the points 0, 1, 2, 3, ... are limit points.
Answer. All positive numbers are limit points.
Question. The limit points of the sequence a_n = (-1)^n n / (2n+1) for n = 0, 1, 2, 3, ... are
Answer. Just 1/2
Answer. 0, -1/3, 2/5, -3/7, ...
Answer. 0 and 1/2
Correct Answer. -1/2 and 1/2
Answer. 1 and -1
Answer. 1/2 and 1
Answer. No limit points exist.
Question. The inf of the sequence a_n = 1 + (-1)^n / n for n = 1,2,3,... is
Correct Answer. 0
Answer. 1
Answer. 3/2
Answer. 0 and 1
Answer. Does not exist
Answer. Negative infinity
Question. The least upper bound of the set { x in R: x^2 < 2 } is
Correct Answer. sqrt(2)
Answer. 0
Answer. 2
Answer. 4
Answer. Positive infinity
Answer. Negative infinity
Answer. Does not exist
Question. The supremum of the set { x in R: x^2 > 2 } is
Answer. sqrt(2)
Answer. 0
Answer. 2
Answer. 4
Correct Answer. Positive infinity
Answer. Negative infinity
Partially Correct Answer. Does not exist
Comment. 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 +infinity and -infinity).
Question. The infimum of the set { x in R: x^2 <= 2 } is
Answer. sqrt(2)
Correct Answer. -sqrt(2)
Answer. 0
Answer. 4
Answer. Positive infinity
Answer. Negative infinity
Answer. Does not exist
Question. The least upper bound of the set { x^2 in R: x <= 2 } is
Answer. sqrt(2)
Answer. 0
Answer. 2
Answer. 4
Correct Answer. Positive infinity
Answer. Negative infinity
Partially Correct Answer. Does not exist
Comment. 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 +infinity and -infinity).
Question. The infimum of the set { x^2 in R: x > 2 } is
Answer. sqrt(2)
Answer. 0
Answer. 2
Correct Answer. 4
Answer. Positive infinity
Answer. Negative infinity
Answer. Does not exist
Question. The infimum of the empty set is
Answer. 0
Answer. An arbitrary real number
Answer. The empty set
Correct Answer. Positive infinity
Comment. It may seem unintuitive, but every real number is a lower bound for the empty set, and thus the greatest lower bound (infimum) is +infinity.
Answer. Negative infinity
Answer. Does not exist
Question. The supremum of the empty set is
Answer. 0
Answer. An arbitrary real number
Answer. The empty set
Correct Answer. Negative infinity
Comment. It may seem unintuitive, but every real number is an upper bound for the empty set, and thus the least upper bound (supremum) is -infinity.
Answer. Positive infinity
Answer. Does not exist
Question. Let a_1, a_2, ... be a sequence of real numbers. What does it mean for a_n to converge to a limit c as n tends to infinity?
Correct Answer. For every eps > 0, there exists an N >= 1 such that |a_n - c| <= eps for all n >= N.
Answer. The sequence |a_n-c| is decreasing, i.e. |a_{n+1}-c| < |a_n-c| for all n.
Answer. For every N >= 1, there exists an eps > 0 such that |a_n - c| <= eps for all n >= N.
Answer. For every N >= 1, and every eps > 0, we have |a_n - c| <= eps for all n >= N.
Answer. There exists N >= 1 such that for every eps > 0, we have |a_n - c| <= eps for all n >= N.
Answer. There exists N >= 1 and there exists eps > 0 such that |a_n - c| <= eps for all n >= N.
Answer. For every eps > 0, there exists an N >= 1 such that |a_n - c| <= eps for infinitely many n >= N.
Comment. This is what it means for c to be a limit point of a_n.
Question. Let a_1, a_2, ... be a sequence of real numbers. What does it mean for the sequence a_n to have c as a limit point?
Answer. For every eps > 0, there exists an N >= 1 such that |a_n - c| <= eps for all n >= N.
Comment. This is what it means for a_n to converge to c, which is a slightly different concept.
Answer. For every N >= 1, there exists an eps > 0 and n >= N such that |a_n - c| <= eps.
Answer. For every N >= 1, and every eps > 0, there exists n >= N such that |a_n - c| <= eps.
Answer. There exists N >= 1 such that for every eps > 0, there exists n >= N such that |a_n - c| <= eps.
Answer. There exists N >= 1 and there exists eps > 0 such that |a_n - c| <= eps for all n >= N.
Correct Answer. For every eps > 0 and N >= 1, there exists n >= N such that |a_n - c| <= eps.
Question. Let a_1, a_2, ... be a sequence of real numbers. What does it mean for the sequence a_n to be bounded?
Correct Answer. There exists an M >= 0 such that |a_n| <= M for all n >= 1.
Answer. There exists an M >= 0 such that |a_n| <= M for some n >= 1.
Answer. For each n >= 1 there exists an M >= 0 such that |a_n| <= M.
Answer. For every n >= 1 and every M >= 0 we have |a_n| <= M.
Answer. There exists an M >= 0 such that a_n < M or a_n > -M for each n >= 1.
Answer. There exists an n >= 1 and there exists M >= 0 such that |a_n| <= M.
Answer. The sequence a_1,a_2,... either has a lower bound M, or an upper bound M, or both.
Comment. One needs _both_ an upper bound and a lower bound in order to be bounded.
Question. Let a_1, a_2, ... be a sequence of real numbers. What does it mean for a_n to be a Cauchy sequence?
Correct Answer. For every eps > 0, there exists an N >= 1 such that |a_n - a_m| <= eps for all n,m >= N.
Answer. The sequence |a_n-a_{n+1}| is decreasing, i.e. |a_{n+1}-a_{n+2}| < |a_n-a_{n+1}| for all n.
Answer. For every N >= 1, there exists an eps > 0 such that |a_n - a_m| <= eps for all n,m >= N.
Answer. For every N >= 1, and every eps > 0, we have |a_n - a_m| <= eps for all n,m >= N.
Answer. There exists N >= 1 such that for every eps > 0, we have |a_n - a_m| <= eps for all n,m >= N.
Answer. There exists N >= 1 and there exists eps > 0 such that |a_n - a_m| <= eps for all n,m >= N.
Answer. For every eps > 0, there exists an N >= 1 such that |a_n - a_m| <= eps for infinitely many n,m >= N.
Question. Let E be a subset of the reals R. What does it mean when we say that x is the supremum of E?
Correct Answer. x is an upper bound for E, and for any other upper bound x' of E we have x <= x'.
Answer. x is the element in E which is larger than all the others.
Answer. x is the lim sup of every sequence in E.
Answer. For every y in E, we have x > y.
Answer. For every y in E, we have x >= y.
Comment. This says that x is an upper bound for E, but does not say that it is the least upper bound for E.
Answer. x is an upper bound for E, and is larger than all other upper bounds for E.
Answer. Every number less than x lies in E, and every number greater than x lies outside of E.
Question. Let E be a subset of the reals R. What does it mean when we say that x is the infimum of E?
Correct Answer. x is a lower bound for E, and is larger than all other lower bounds for E.
Answer. x is a lower bound for E, and is smaller than all other lower bounds for E.
Answer. x is the element in E which is smaller than all the others.
Answer. x is the lim inf of every sequence in E.
Answer. For every y in E, we have y > x.
Answer. For every y in E, we have y >= x.
Comment. This says that x is an lower bound for E, but does not say that it is the greatest lower bound for E.
Answer. Every number greater than x lies in E, and every number less than x lies outside of E.