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

TBIL Community

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Section 8.3 Partial Sum Sequence (SQ3)

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Subsection 8.3.1 Activities

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Activity 8.3.1.

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Consider the sequence {an}n=0∞={12n}n=0∞.
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(a)
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Find the first 5 terms of this sequence.
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(b)
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Compute the following:
  1. a0.
  2. a0+a1.
  3. a0+a1+a2.
  4. a0+a1+a2+a3.
  5. a0+a1+a2+a3+a4.
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Activity 8.3.2.

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Consider the sequence {an}n=1∞={1n}n=1∞.
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(a)
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Find the first 5 terms of this sequence.
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(b)
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Compute the following:
  1. a1.
  2. a1+a2.
  3. a1+a2+a3.
  4. a1+a2+a3+a4.
  5. a1+a2+a3+a4+a5.
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Definition 8.3.3.

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Given a sequence {an}n=0∞ define the kth partial sum for this sequence to be
Ak=∑i=0kai=a0+a1+a2+⋯+ak.
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Note that {An}n=0∞=A0,A1,A2,… is itself a sequence called the partial sum sequence.
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More generally, partial sums may be defined for any starting index. Given {an}n=N∞, let
Ak=∑i=Nkai=aN+aN+1+aN+2+⋯+ak.
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Activity 8.3.5.

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Consider the sequence an=23n. What is the best way to find the 100th partial sum A100?
  1. Sum the first 101 terms of the sequence {an}.
  2. Find a closed form for the partial sum sequence {An}.
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Activity 8.3.6.

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Expand the following polynomial products, and then reduce to as few summands as possible.
  1. (1−x)(1+x+x2).
  2. (1−x)(1+x+x2+x3).
  3. (1−x)(1+x+x2+x3+x4).
  4. (1−x)(1+x+x2+⋯+xn), where n is any nonnegative integer.
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Activity 8.3.7.

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Suppose S5=1+12+14+18+116+132. Without actually computing this sum, which of the following is equal to (1−12)S5?
  1. 12+14+18+116+132−164.
  2. 1−164.
  3. 1−12−14−18−116−132.
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Activity 8.3.8.

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Recall from Activity 8.3.4 that A100=2+23+232+233+234+⋯+23100=2(1+13+132+133+134+⋯+13100).
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(a)
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Which of the following is equal to (1−13)A100?
  1. 1−13101.
  2. 1−13100.
  3. 2(1−13101).
  4. 2(1−13100).
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(b)
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Based on your previous choice, write out an expression for A100.
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Activity 8.3.9.

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Suppose that {bn}n=0∞={(−2)n}n=0∞={1,−2,4,−8,…}. Let Bn=∑i=0nbi be the nth partial sum of {bn}.
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(a)
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Find simple expressions for the following:
  1. (1−(−2))B10.
  2. (1−(−2))B30.
  3. (1−(−2))Bn. Choose from the following:
    1. 1+(−2)n.
    2. 1−(−2)n.
    3. 1+(−2)n+1.
    4. 1−(−2)n+1.
    5. 1−2n.
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(b)
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Based on your previous answers, solve for the following:
  1. B10.
  2. B30.
  3. Bn. Choose from the following:
    1. 1−(−2)n+11−(−2)
    2. 1−(−2)n+11−2
    3. 1−(−2)n+11+(−2)
    4. 1−(−2)n1−2
    5. 1−(−2)n1−(−2)
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Activity 8.3.10.

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Consider the following sequences:
  1. {an}n=0∞={(−23)n}n=0∞.
  2. {bn}n=0∞={2⋅(−1)n}n=0∞.
  3. {cn}n=0∞={−3⋅(1.2)n}n=0∞.
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(a)
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Find the closed form for the nth partial sum for the geometric sequence An=∑i=0nai=∑i=0n(−23)n.
  1. 35(1−(−23)n+1).
  2. 53(1−(−23)n+1).
  3. 53(1+23(23)n).
  4. 35(1+23(23)n).
  5. 1−(−23)n+1.
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(b)
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Find the closed form for the nth partial sum for the geometric sequence Bn=∑i=0nbi=∑i=0n2⋅(−1)n.
  1. 2n+1.
  2. 1−(−1)n+1.
  3. 1+(−1)n.
  4. 2(1+(−1)n).
  5. 2(1−(−1)n+1).
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(c)
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Find the closed form for the nth partial sum for the geometric sequence Cn=∑i=0nci=∑i=0n−3⋅(1.2)n.
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Activity 8.3.11.

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Given the closed forms you found in Activity 8.3.10, which of the following limits are defined? If defined, what is the limit?
  1. limn→∞An.
  2. limn→∞Bn.
  3. limn→∞Cn.
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Definition 8.3.12.

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Given a sequence an, we define the limit of the series
∑n=k∞an:=limn→∞An
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where An=∑i=knai. We call ∑n=k∞an an infinite series.
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Activity 8.3.13.

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Which of the following series are infinite?
  1. ∑n=0∞3(0.8)n.
  2. ∑n=0∞2(54)n.
  3. ∑n=0∞(56)n.
  4. ∑n=0∞12(81)n.
  5. ∑n=0∞10(−15)n.
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Activity 8.3.14.

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Let {an}n=1∞={1n−1n+1}=1−12,12−13,13−14,…. Let An=∑i=1nai=∑i=1n(1i−1i+1).
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Which of the following is the best strategy for evaluating A4=(1−12)+(12−13)+(13−14)+(14−15)?
  1. Compute A4=(1−12)+(12−13)+(13−14)+(14−15)=12+16+112+120, then evaluate the sum.
  2. Rewrite A4=(1−12)+(12−13)+(13−14)+(14−15)=1+(−12+12)+(−13+13)+(−14+14)−15, then simplify.
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Activity 8.3.15.

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Recall from Activity 8.3.14 that {an}n=1∞={1n−1n+1} and An=∑i=1nai=∑i=1n(1i−1i+1).
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Compute the following partial sums:
  1. A3.
  2. A10.
  3. A100.
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Activity 8.3.16.

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Recall from Activity 8.3.14 that {an}n=1∞={1n−1n+1} and An=∑i=1nai=∑i=1n(1i−1i+1).
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Which of the following is equal to An?
  1. n−1n+1.
  2. 1−1n.
  3. 1−1n+1.
  4. 1−1i.
  5. 1−1i+1.
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Definition 8.3.17.

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Given a sequence {xn}1∞ and a sequence of the form {sn}1∞:={xn−xn+1}1∞ we call the series Sn=∑i=1nsi=∑i=1n(xi−xi+1) to be a telescoping series.
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Activity 8.3.18.

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Given a telescoping series Sn=∑i=1nsi=∑i=1n(xi−xi+1), find:
  1. S2.
  2. S10.
  3. Choose Sn from the following options:
    1. x1−xn
    2. x1−xn+1
    3. x1−xn−1
    4. x1−xn+1
    5. x1−xn−1
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Activity 8.3.19.

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For each of the following telescoping series, find the closed form for the nth partial sum.
  1. Sn=∑i=1n(2−i−(2−i−1)).
  2. Sn=∑i=1n(i2−(i+1)2).
  3. Sn=∑i=1n(12i+1−12i+3).
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Activity 8.3.20.

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Given the closed forms you found in Activity 8.3.19, determine which of the following telescoping series converge. If so, to what value does it converge?
  1. ∑i=1∞(2−i−(2−i−1)).
  2. ∑i=1∞(i2−(i+1)2).
  3. ∑i=1∞(12i+1−12i+3).
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Activity 8.3.21.

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Consider the partial sum sequence An=(−2)+(23)+(−29)+⋯+(−2⋅(−13)n).
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(a)
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Find a closed form for An.
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(b)
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Does {An} converge? If so, to what value?
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Activity 8.3.22.

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Consider the partial sum sequence Bn=∑i=1n(15i+2−15i+7).
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(a)
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Find a closed form for Bn.
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(b)
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Does {Bn} converge? If so, to what value?
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Subsection 8.3.2 Videos

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Figure 178. Video: Compute the first few terms of a telescoping or geometric partial sum sequence, and find a closed form for this sequence, and compute its limit.
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Subsection 8.3.3 Exercises

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