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Calculus for Team-Based Inquiry Learning: 2023 Edition

TBIL Fellows

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Section 8.2 Sequence Properties and Limits (SQ2)

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Subsection 8.2.1 Activities

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Activity 8.2.1.

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We will consider the function f(x)=4x+8x.
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(b)
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Determine on which intervals f(x) is increasing and/or decreasing. (Hint: compute f′(x) first.)
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(c)
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Which statement best describes f(x) for x>0?
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    f(x) is bounded above by 4
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    f(x) is bounded below by 4
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    f(x) is bounded above and below by 4
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    f(x) is not bounded above
  5. 🔗
    f(x) is not bounded below
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Definition 8.2.1.

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Given a sequence {xn}:
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    {xn} is monotonically increasing if xn+1>xn for every choice of n.
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    {xn} is monotonically non-decreasing if xn+1≥xn for every choice of n.
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    {xn} is monotonically decreasing if xn+1<xn for every choice of n.
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    {xn} is monotonically non-increasing if xn+1≤xn for every choice of n.
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All of these sequences would be monotonic.
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Activity 8.2.2.

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Consider the sequence {(−1)nn}n=1∞.
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(b)
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Which of the following is true about xn+1−xn? There can be more or less than one answer.
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    xn+1−xn>0 for every choice of n.
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    xn+1−xn≥0 for every choice of n.
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    xn+1−xn<0 for every choice of n.
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    xn+1−xn≤0 for every choice of n.
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(c)
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Which of the following (if any) describe {(−1)nn}n=1∞?
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    Monotonically increasing.
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    Monotonically non-decreasing.
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    Monotonically decreasing.
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    Monotonically non-increasing.
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Activity 8.2.3.

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Consider the sequence {n2+1n}n=1∞.
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(b)
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Which of the following is true about xn+1−xn? There can be more or less than one answer.
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    xn+1−xn>0 for every choice of n.
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    xn+1−xn≥0 for every choice of n.
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    xn+1−xn<0 for every choice of n.
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    xn+1−xn≤0 for every choice of n.
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(c)
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Which of the following (if any) describe {n2+1n}n=1∞?
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    Monotonically increasing.
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    Monotonically non-decreasing.
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    Monotonically decreasing.
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    Monotonically non-increasing.
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Activity 8.2.4.

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

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Consider the sequence {23n}n=0∞.
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(b)
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Which of the following is true about xn+1−xn? There can be more or less than one answer.
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    xn+1−xn>0 for every choice of n.
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    xn+1−xn≥0 for every choice of n.
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    xn+1−xn<0 for every choice of n.
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    xn+1−xn≤0 for every choice of n.
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(c)
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Which of the following (if any) describe {23n}n=0∞?
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    Monotonically increasing.
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    Monotonically non-decreasing.
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    Monotonically decreasing.
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    Monotonically non-increasing.
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Definition 8.2.2.

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A sequence {xn} is bounded if there are real numbers bu,bℓ such that
bℓ≤xn≤bu
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for every n.
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Activity 8.2.6.

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Consider the sequence {(−1)nn}n=1∞ from Activity 8.2.2.
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(a)
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Is there a bu such that xn≤bu for every n? If so, what would be one such bu?
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(b)
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Is there a bℓ such that bℓ≤xn for every n? If so, what would be one such bℓ?
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(c)
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Is {(−1)nn}n=1∞ bounded?
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Activity 8.2.7.

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Consider the sequence {n2+1n}n=1∞ from Activity 8.2.3.
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(a)
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Is there a bu such that xn≤bu for every n? If so, what would be one such bu?
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(b)
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Is there a bℓ such that bℓ≤xn for every n? If so, what would be one such bℓ?
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Activity 8.2.8.

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Consider the sequence {n+1n}n=1∞ from Activity 8.2.4.
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(a)
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Is there a bu such that xn≤bu for every n? If so, what would be one such bu?
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(b)
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Is there a bℓ such that bℓ≤xn for every n? If so, what would be one such bℓ?
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Activity 8.2.9.

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Consider the sequence {23n}n=1∞ from Activity 8.2.5.
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(a)
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Is there a bu such that xn≤bu for every n? If so, what would be one such bu?
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(b)
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Is there a bℓ such that bℓ≤xn for every n? If so, what would be one such bℓ?
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Definition 8.2.3.

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Given a sequence {xn}, we say xn has limit L, denoted
limn→∞xn=L
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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} does not converge.
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Activity 8.2.10.

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(a)
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For each of the following, determine if the sequence converges.
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    {(−1)nn}n=1∞.
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    {n2+1n}n=1∞.
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    {n+1n}n=1∞.
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    {23n}n=0∞.
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(b)
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Where possible, find the limit of the sequence.
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Activity 8.2.11.

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(a)
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Determine to what value {4nn+1}n=0∞ converges.
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(b)
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Which of the following ia most likely true about {4n(−1)nn+1}n=0∞?
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    {4n(−1)nn+1}n=0∞ converges to 4.
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    {4n(−1)nn+1}n=0∞ converges to 0.
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    {4n(−1)nn+1}n=0∞ converges to -4.
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    {4n(−1)nn+1}n=0∞ does not converge.
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Activity 8.2.12.

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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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(b)
{n33n}n=0∞.
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(c)
{nn+3}n=1∞.
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(d)
{(−1)nn+3}n=1∞.
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Subsection 8.2.2 Videos

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Figure 170. Video: Determine if a sequence is convergent, divergent, monotonic, or bounded, and compute limits of convergent sequences
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