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Sequences

This quiz covers the basic concepts of sequences, including definitions, properties, and operations.

14 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is a sequence?

  1. A function whose domain is the set of natural numbers
  2. A function whose range is the set of natural numbers
  3. A function whose domain and range are both the set of natural numbers
  4. None of the above
Question 2 Multiple Choice (Single Answer)

What is the limit of a sequence?

  1. The value that the sequence approaches as the index approaches infinity
  2. The value that the sequence takes at infinity
  3. The value that the sequence takes at the origin
  4. None of the above
Question 3 Multiple Choice (Single Answer)

What is a convergent sequence?

  1. A sequence that has a limit
  2. A sequence that does not have a limit
  3. A sequence that is increasing
  4. A sequence that is decreasing
Question 4 Multiple Choice (Single Answer)

What is a divergent sequence?

  1. A sequence that has a limit
  2. A sequence that does not have a limit
  3. A sequence that is increasing
  4. A sequence that is decreasing
Question 5 Multiple Choice (Single Answer)

What is the Cauchy criterion for convergence?

  1. If the limit of the sequence exists, then the sequence is Cauchy.
  2. If the sequence is Cauchy, then the limit of the sequence exists.
  3. Both of the above
  4. None of the above
Question 6 Multiple Choice (Single Answer)

What is the ratio test for convergence?

  1. If the limit of the ratio of consecutive terms of the sequence is less than 1, then the sequence is convergent.
  2. If the limit of the ratio of consecutive terms of the sequence is greater than 1, then the sequence is divergent.
  3. If the limit of the ratio of consecutive terms of the sequence is equal to 1, then the sequence may be convergent or divergent.
  4. None of the above
Question 7 Multiple Choice (Single Answer)

What is the root test for convergence?

  1. If the limit of the nth root of the absolute value of the nth term of the sequence is less than 1, then the sequence is convergent.
  2. If the limit of the nth root of the absolute value of the nth term of the sequence is greater than 1, then the sequence is divergent.
  3. If the limit of the nth root of the absolute value of the nth term of the sequence is equal to 1, then the sequence may be convergent or divergent.
  4. None of the above
Question 8 Multiple Choice (Single Answer)

What is the comparison test for convergence?

  1. If the sequence is bounded above by a convergent sequence, then the sequence is convergent.
  2. If the sequence is bounded below by a divergent sequence, then the sequence is divergent.
  3. Both of the above
  4. None of the above
Question 9 Multiple Choice (Single Answer)

What is the limit comparison test for convergence?

  1. If the limit of the ratio of the nth term of the sequence to the nth term of a convergent sequence is a positive number, then the sequence is convergent.
  2. If the limit of the ratio of the nth term of the sequence to the nth term of a divergent sequence is a positive number, then the sequence is divergent.
  3. Both of the above
  4. None of the above
Question 10 Multiple Choice (Single Answer)

What is the alternating series test for convergence?

  1. If the terms of the sequence are alternately positive and negative, and the absolute value of the terms is decreasing, then the sequence is convergent.
  2. If the terms of the sequence are alternately positive and negative, and the absolute value of the terms is increasing, then the sequence is divergent.
  3. Both of the above
  4. None of the above
Question 11 Multiple Choice (Single Answer)

What is the absolute convergence test?

  1. If the sequence of absolute values of the terms of the sequence is convergent, then the sequence is absolutely convergent.
  2. If the sequence of absolute values of the terms of the sequence is divergent, then the sequence is conditionally convergent.
  3. Both of the above
  4. None of the above
Question 12 Multiple Choice (Single Answer)

What is the conditional convergence test?

  1. If the sequence of absolute values of the terms of the sequence is divergent, then the sequence is conditionally convergent.
  2. If the sequence of absolute values of the terms of the sequence is convergent, then the sequence is absolutely convergent.
  3. Both of the above
  4. None of the above
Question 13 Multiple Choice (Single Answer)

What is the squeeze theorem?

  1. If two sequences are both convergent to the same limit, and a third sequence is bounded above by the first sequence and bounded below by the second sequence, then the third sequence is convergent to the same limit.
  2. If two sequences are both divergent to infinity, and a third sequence is bounded above by the first sequence and bounded below by the second sequence, then the third sequence is divergent to infinity.
  3. Both of the above
  4. None of the above
Question 14 Multiple Choice (Single Answer)

What is the monotonic sequence theorem?

  1. A monotonic sequence that is bounded is convergent.
  2. A monotonic sequence that is unbounded is divergent.
  3. Both of the above
  4. None of the above
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