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Sequences

Description: This quiz covers the basic concepts of sequences, including definitions, properties, and operations.
Number of Questions: 14
Created by:
Tags: sequences limits convergence divergence
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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

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Correct Option: A
Explanation:

A sequence is a function whose domain is the set of natural numbers.

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

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Correct Option: A
Explanation:

The limit of a sequence is the value that the sequence approaches as the index approaches infinity.

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

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Correct Option: A
Explanation:

A convergent sequence is a sequence that has a limit.

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

Show answer
Correct Option: B
Explanation:

A divergent sequence is a sequence that does not have a limit.

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

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Correct Option: C
Explanation:

The Cauchy criterion for convergence states that if the limit of the sequence exists, then the sequence is Cauchy, and if the sequence is Cauchy, then the limit of the sequence exists.

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

Show answer
Correct Option: A
Explanation:

The ratio test for convergence states that if the limit of the ratio of consecutive terms of the sequence is less than 1, then the sequence is convergent.

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

Show answer
Correct Option: A
Explanation:

The root test for convergence states that 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.

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

Show answer
Correct Option: C
Explanation:

The comparison test for convergence states that if the sequence is bounded above by a convergent sequence, then the sequence is convergent, and if the sequence is bounded below by a divergent sequence, then the sequence is divergent.

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

Show answer
Correct Option: C
Explanation:

The limit comparison test for convergence states that 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, and 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.

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

Show answer
Correct Option: A
Explanation:

The alternating series test for convergence states that 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.

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

Show answer
Correct Option: A
Explanation:

The absolute convergence test states that if the sequence of absolute values of the terms of the sequence is convergent, then the sequence is absolutely convergent.

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

Show answer
Correct Option: A
Explanation:

The conditional convergence test states that if the sequence of absolute values of the terms of the sequence is divergent, then the sequence is conditionally convergent.

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

Show answer
Correct Option: A
Explanation:

The squeeze theorem states that 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.

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

Show answer
Correct Option: C
Explanation:

The monotonic sequence theorem states that a monotonic sequence that is bounded is convergent, and a monotonic sequence that is unbounded is divergent.

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