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Calculus for Team-Based Inquiry Learning: PREVIEW Edition — Instructor Version

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Section 8.2 Sequence Properties and Limits (SQ2)

Subsection 8.2.1 Activities

Activity 8.2.1.

We will consider the function f(x)=4x+8x.
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(a)
Compute the limit limx→∞4x+8x.
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  1. 0
  2. 8
  3. 1
  4. 4
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(b)
Determine on which intervals f(x) is increasing and/or decreasing. (Hint: compute f′(x) first.)
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(c)
Which statement best describes f(x) for x>0?
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  1. f(x) is bounded above by 4
  2. f(x) is bounded below by 4
  3. f(x) is bounded above and below by 4
  4. f(x) is not bounded above
  5. f(x) is not bounded below
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Definition 8.2.2.

Given a sequence {xn}:
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  • {xn} is monotonically increasing if xn+1>xn for every choice of n.
  • {xn} is monotonically non-decreasing if xn+1≥xn for every choice of n.
  • {xn} is monotonically decreasing if xn+1<xn for every choice of n.
  • {xn} is monotonically non-increasing if xn+1≤xn for every choice of n.
All of these sequences would be monotonic.
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Activity 8.2.3.

Consider the sequence {(−1)nn}n=1∞.
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(b)
Which of the following is true about xn+1−xn? There can be more or less than one answer.
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  1. xn+1−xn>0 for every choice of n.
  2. xn+1−xn≥0 for every choice of n.
  3. xn+1−xn<0 for every choice of n.
  4. xn+1−xn≤0 for every choice of n.
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(c)
Which of the following (if any) describe {(−1)nn}n=1∞?
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  1. Monotonically increasing.
  2. Monotonically non-decreasing.
  3. Monotonically decreasing.
  4. Monotonically non-increasing.
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Activity 8.2.4.

Consider the sequence {n2+1n}n=1∞.
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(b)
Which of the following is true about xn+1−xn? There can be more or less than one answer.
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  1. xn+1−xn>0 for every choice of n.
  2. xn+1−xn≥0 for every choice of n.
  3. xn+1−xn<0 for every choice of n.
  4. xn+1−xn≤0 for every choice of n.
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(c)
Which of the following (if any) describe {n2+1n}n=1∞?
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  1. Monotonically increasing.
  2. Monotonically non-decreasing.
  3. Monotonically decreasing.
  4. Monotonically non-increasing.
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Activity 8.2.5.

Consider the sequence {n+1n}n=1∞.
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(b)
Which of the following is true about xn+1−xn? There can be more or less than one answer.
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  1. xn+1−xn>0 for every choice of n.
  2. xn+1−xn≥0 for every choice of n.
  3. xn+1−xn<0 for every choice of n.
  4. xn+1−xn≤0 for every choice of n.
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(c)
Which of the following (if any) describe {n+1n}n=1∞?
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  1. Monotonically increasing.
  2. Monotonically non-decreasing.
  3. Monotonically decreasing.
  4. Monotonically non-increasing.
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Activity 8.2.6.

Consider the sequence {23n}n=0∞.
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(b)
Which of the following is true about xn+1−xn? There can be more or less than one answer.
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  1. xn+1−xn>0 for every choice of n.
  2. xn+1−xn≥0 for every choice of n.
  3. xn+1−xn<0 for every choice of n.
  4. xn+1−xn≤0 for every choice of n.
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(c)
Which of the following (if any) describe {23n}n=0∞?
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  1. Monotonically increasing.
  2. Monotonically non-decreasing.
  3. Monotonically decreasing.
  4. Monotonically non-increasing.
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Definition 8.2.7.

A sequence {xn} is bounded if there are real numbers bu,bℓ such that
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bℓ≤xn≤bu
for every n.
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Activity 8.2.8.

Consider the sequence {(−1)nn}n=1∞ from Activity 8.2.3.
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(a)
Is there a bu such that xn≤bu for every n? If so, what would be one such bu?
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(b)
Is there a bℓ such that bℓ≤xn for every n? If so, what would be one such bℓ?
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(c)
Is {(−1)nn}n=1∞ bounded?
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Activity 8.2.9.

Consider the sequence {n2+1n}n=1∞ from Activity 8.2.4.
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(a)
Is there a bu such that xn≤bu for every n? If so, what would be one such bu?
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(b)
Is there a bℓ such that bℓ≤xn for every n? If so, what would be one such bℓ?
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Activity 8.2.10.

Consider the sequence {n+1n}n=1∞ from Activity 8.2.5.
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(a)
Is there a bu such that xn≤bu for every n? If so, what would be one such bu?
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(b)
Is there a bℓ such that bℓ≤xn for every n? If so, what would be one such bℓ?
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Activity 8.2.11.

Consider the sequence {23n}n=1∞ from Activity 8.2.6.
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(a)
Is there a bu such that xn≤bu for every n? If so, what would be one such bu?
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(b)
Is there a bℓ such that bℓ≤xn for every n? If so, what would be one such bℓ?
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Definition 8.2.12.

Given a sequence {xn}, we say xn has limit L, denoted
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limn→∞xn=L
if we can make xn as close to L as we like by making n sufficiently large. If such an L exists, we say {xn} converges to L. If no such L exists, we say {xn} diverges.
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Activity 8.2.13.

(a)
For each of the following, determine if the sequence converges.
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  1. {(−1)nn}n=1∞.
  2. {n2+1n}n=1∞.
  3. {n+1n}n=1∞.
  4. {23n}n=0∞.
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(b)
Where possible, find the limit of the sequence.
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Activity 8.2.14.

(a)
Determine to what value {4nn+1}n=0∞ converges.
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(b)
Which of the following is most likely true about {4n(−1)nn+1}n=0∞?
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  1. {4n(−1)nn+1}n=0∞ converges to 4.
  2. {4n(−1)nn+1}n=0∞ converges to 0.
  3. {4n(−1)nn+1}n=0∞ converges to -4.
  4. {4n(−1)nn+1}n=0∞ does not converge.
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Activity 8.2.15.

For each of the following sequences, determine which of the properties: monotonic, bounded and convergent, the sequence satisfies. If a sequence is convergent, determine to what it converges.
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Subsection 8.2.2 Videos

Figure 182. Video: Determine if a sequence is convergent, divergent, monotonic, or bounded, and compute limits of convergent sequences
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Subsection 8.2.3 Exercises

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