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Quiz: Sequences
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Saved by RH
on December 21, 2008 at 11:28:16 am
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.)
- In the rationals , the sequence 1, 1/2, 1/3, 1/4, 1/5, 1/6, ... is
- Bounded.
- Convergent.
- A Cauchy sequence.
- A and C.
- B and C.
- A, B, and C.
- None of the above.
- In the rationals , the sequence 1, -1, 1, -1, 1, -1, ... is
- Bounded.
- Convergent.
- A Cauchy sequence.
- A and C.
- B and C.
- A, B, and C.
- None of the above.
- In the reals , the sequence 1, -1/2, 1/3, -1/4, 1/5, -1/6, ... is
- Bounded.
- Convergent.
- A Cauchy sequence.
- A and C.
- B and C.
- A, B, and C.
- None of the above.
- In the rationals , the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... is
- Bounded.
- Convergent.
- A Cauchy sequence.
- A and C.
- B and C.
- A, B, and C.
- None of the above.
- In the reals , the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... is
- Bounded.
- Convergent.
- A Cauchy sequence.
- A and C.
- B and C.
- A, B, and C.
- None of the above.
- In the rationals , the sequence 1, -2, 3, -4, 5, -6, ...is
- Bounded.
- Convergent.
- A Cauchy sequence.
- A and C.
- B and C.
- A, B, and C.
- None of the above.
- In the rationals , the sequence 0.9, -0.99, 0.999, -0.9999, 0.99999, -0.999999, ... is
- Bounded.
- Convergent.
- A Cauchy sequence.
- A and C.
- B and C.
- A, B, and C.
- None of the above.
- The lim sup of the sequence 1.1, -1.01, 1.001, -1.0001, 1.00001, ... is
- 1
- 1.1
- +1 and -1
- -1
- Does not exist
- The lim inf of the sequence 0, -1, 0, -2, 0, -3, 0, -4, 0, -5, ... is
-
- 0
- -1
- -5
- Does not exist
- The sup of the sequence 1, -1/2, 1/3, -1/4, 1/5, -1/6, ... is
- 1
- 0
- -1/2
- Does not exist
- The limit points of the sequence 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ... are
- Just 0
- 0 and
- No limit points exist.
- All the points 0, 1, 2, 3, ... are limit points.
- All positive numbers are limit points.
- The limit points of the sequence for are
- Just 1/2
- 0, -1/3, 2/5, -3/7, ...
- 0 and 1/2
- -1/2 and 1/2
- 1 and -1
- 1/2 and 1
- No limit points exist.
- The inf of the sequence for is
- 0
- 1
- 3/2
- 0 and 1
- Does not exist
- The least upper bound of the set is
-
- 0
- 2
- 4
- Does not exist
- The supremum of the set is
- 0
- 2
- 4
-
- 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 and ).
- The infimum of the set is
-
- 0
- 4
- Does not exist
- The least upper bound of the set is
- 0
- 2
- 4
-
- 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 and ).
- The infimum of the set is
- 0
- 2
- 4
- Does not exist
- The infimum of the empty set is
- 0
- An arbitrary real number
- The empty set
- 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 .
- Does not exist
- The supremum of the empty set is
- 0
- An arbitrary real number
- The empty set
- 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 .
- Does not exist
- Let be a sequence of real numbers. What does it mean for to converge to a limit as ?
- For every , there exists an such that for all .
- The sequence is decreasing, i.e. for all .
- For every , there exists an such that for all .
- For every , and every , we have for all .
- There exists such that for every , we have for all .
- There exists and there exists such that for all .
- For every , there exists an such that for infinitely many .
- INCORRECT. This is what it means for to be a limit point of .
- Let be a sequence of real numbers. What does it mean for the sequence to have as a limit point?
- For every , there exists an such that for all .
- INCORRECT. This is what it means for to converge to , which is a slightly different concept.
- For every , there exists an and such that .
- For every , and every , there exists such that .
- There exists such that for every , there exists such that .
- There exists and there exists such that for all .
- For every and , there exists such that .
- Let be a sequence of real numbers. What does it mean for the sequence to be bounded?
- There exists an such that for all .
- There exists an such that for some .
- For each there exists an such that .
- For every and every we have .
- There exists an such that or for each .
- There exists an and there exists such that .
- The sequence either has a lower bound , or an upper bound , or both.
- INCORRECT. One needs _both_ an upper bound and a lower bound in order to be bounded.
- Let be a sequence of real numbers. What does it mean for to be a Cauchy sequence?
- For every , there exists an such that for all .
- The sequence is decreasing, i.e. for all .
- For every , there exists an such that for all .
- For every , and every , we have n,m \ge N$.
- There exists such that for every , we have for all .
- There exists and there exists such that for all .
- For every , there exists an such that for infinitely many .
- Let be a subset of the reals . What does it mean when we say that is the supremum of ?
- is an upper bound for , and for any other upper bound of we have .
- is the element in which is larger than all the others.
- is the lim sup of every sequence in .
- For every , we have .
- For every , we have .
- INCORRECT. This says that x is an upper bound for , but does not say that it is the least upper bound for .
- is an upper bound for , and is larger than all other upper bounds for .
- Every number less than lies in , and every number greater than lies outside of .
- Let be a subset of the reals . What does it mean when we say that is the infimum of ?
- is a lower bound for , and is larger than all other lower bounds for .
- is a lower bound for , and is smaller than all other lower bounds for .
- is the element in which is smaller than all the others.
- is the lim inf of every sequence in .
- For every , we have .
- For every , we have .
- INCORRECT. This says that is an lower bound for , but does not say that it is the greatest lower bound for .
- Every number greater than lies in , and every number less than lies outside of .
Score:
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Quiz: Sequences
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