TBIL-Calc Partial Sum Sequence (SQ3)

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Section 8.3 Partial Sum Sequence (SQ3)

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Learning Outcomes

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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Activity 8.3.1.

Consider the sequence ${a_{n}}_{n = 0}^{\infty} = {\frac{1}{2^{n}}}_{n = 0}^{\infty} .$

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(a)

Find the first 5 terms of this sequence.

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(b)

Compute the following:

$a_{0} .$
$a_{0} + a_{1} .$
$a_{0} + a_{1} + a_{2} .$
$a_{0} + a_{1} + a_{2} + a_{3} .$
$a_{0} + a_{1} + a_{2} + a_{3} + a_{4} .$

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Activity 8.3.2.

Consider the sequence ${a_{n}}_{n = 1}^{\infty} = {\frac{1}{n}}_{n = 1}^{\infty} .$

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(a)

Find the first 5 terms of this sequence.

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(b)

Compute the following:

$a_{1} .$
$a_{1} + a_{2} .$
$a_{1} + a_{2} + a_{3} .$
$a_{1} + a_{2} + a_{3} + a_{4} .$
$a_{1} + a_{2} + a_{3} + a_{4} + a_{5} .$

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Definition 8.3.3.

Given a sequence ${a_{n}}_{n = 0}^{\infty}$ define the $k^{th}$ partial sum of ${a_{n}}$ to be

A_{k} = \sum_{i = 0}^{k} a_{i} = a_{1} + a_{2} + \dots + a_{k} .

Note that ${A_{n}} = A_{0}, A_{1}, A_{2}, \dots$ is itself a sequence called the partial sum sequence.

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Activity 8.3.4.

Let $a_{n} = \frac{2}{3^{n}} .$ Find the following partial sums:

$A_{0} .$
$A_{1} .$
$A_{2} .$
$A_{3} .$
$A_{100} .$

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Activity 8.3.5.

Consider the sequence $a_{n} = \frac{2}{3^{n}} .$ What is the best way to find the 100th partial sum $A_{100} ?$

Sum the first 101 terms of the sequence ${a_{n}} .$
Find a closed form for the partial sum sequence ${A_{n}} .$

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Activity 8.3.6.

Expand the following polynomial products, and then reduce to as few summands as possible.

$(1 - x) (1 + x + x^{2}) .$
$(1 - x) (1 + x + x^{2} + x^{3}) .$
$(1 - x) (1 + x + x^{2} + x^{3} + x^{4}) .$
$(1 - x) (1 + x + x^{2} + \dots + x^{n}),$ where $n$ is any nonnegative integer.

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Activity 8.3.7.

Suppose $S_{5} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} .$ Without actually computing this sum, which of the following is equal to $(1 - \frac{1}{2}) S_{5} ?$

$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} - \frac{1}{64} .$
$1 - \frac{1}{64} .$
$1 - \frac{1}{2} - \frac{1}{4} - \frac{1}{8} - \frac{1}{16} - \frac{1}{32} .$

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Activity 8.3.8.

Recall from Activity 8.3.4 that $A_{100} = 2 + \frac{2}{3} + \frac{2}{3^{2}} + \frac{2}{3^{3}} + \frac{2}{3^{4}} + \dots + \frac{2}{3^{100}} = 2 (1 + \frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \frac{1}{3^{4}} + \dots + \frac{1}{3^{100}}) .$

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(a)

Which of the following is equal to $(1 - \frac{1}{3}) A_{100} ?$

$1 - \frac{1}{3^{101}} .$
$1 - \frac{1}{3^{100}} .$
$2 (1 - \frac{1}{3^{101}}) .$
$2 (1 - \frac{1}{3^{100}}) .$

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(b)

Based on your previous choice, write out an expression for $A_{100} .$

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Activity 8.3.9.

Suppose that ${b_{n}}_{n = 0}^{\infty} = {(- 2)^{n}}_{n = 0}^{\infty} = {1, - 2, 4, - 8, \dots} .$ Let $B_{n} = \sum_{i = 0}^{n} b_{i}$ be the $n$ th partial sum of ${b_{n}} .$

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(a)

Find simple expressions for the following:

$(1 - (- 2)) B_{10} .$
$(1 - (- 2)) B_{30} .$
$(1 - (- 2)) B_{n} .$ Choose from the following:
1. $1 + (- 2)^{n} .$
2. $1 - (- 2)^{n} .$
3. $1 + (- 2)^{n + 1} .$
4. $1 - (- 2)^{n + 1} .$
5. $1 - 2^{n} .$

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(b)

Based on your previous answers, solve for the following:

$B_{10} .$
$B_{30} .$
$B_{n} .$ Choose from the following:
1. $\frac{1 - (- 2)^{n + 1}}{1 - (- 2)}$
2. $\frac{1 - (- 2)^{n + 1}}{1 - 2}$
3. $\frac{1 - (- 2)^{n + 1}}{1 + (- 2)}$
4. $\frac{1 - (- 2)^{n}}{1 - 2}$
5. $\frac{1 - (- 2)^{n}}{1 - (- 2)}$

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Activity 8.3.10.

Consider the following sequences:

${a_{n}}_{n = 0}^{\infty} = {{(- \frac{2}{3})}^{n}}_{n = 0}^{\infty} .$
${b_{n}}_{n = 0}^{\infty} = {2 \cdot {(- 1)}^{n}}_{n = 0}^{\infty} .$
${c_{n}}_{n = 0}^{\infty} = {- 3 \cdot {(1.2)}^{n}}_{n = 0}^{\infty} .$

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(a)

Find the closed form for the $n$ th partial sum for the geometric sequence $A_{n} = \sum_{i = 0}^{n} a_{i} = \sum_{i = 0}^{n} {(- \frac{2}{3})}^{n} .$

$\frac{3}{5} (1 - {(- \frac{2}{3})}^{n + 1}) .$
$\frac{5}{3} (1 - {(- \frac{2}{3})}^{n + 1}) .$
$\frac{5}{3} (1 + \frac{2}{3} {(\frac{2}{3})}^{n}) .$
$\frac{3}{5} (1 + \frac{2}{3} {(\frac{2}{3})}^{n}) .$
$1 - {(- \frac{2}{3})}^{n + 1} .$

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(b)

Find the closed form for the $n$ th partial sum for the geometric sequence $B_{n} = \sum_{i = 0}^{n} b_{i} = \sum_{i = 0}^{n} 2 \cdot {(- 1)}^{n} .$

$2^{n + 1} .$
$1 - (- 1)^{n + 1} .$
$1 + (- 1)^{n} .$
$2 (1 + (- 1)^{n}) .$
$2 (1 - (- 1)^{n + 1}) .$

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(c)

Find the closed form for the $n$ th partial sum for the geometric sequence $C_{n} = \sum_{i = 0}^{n} c_{i} = \sum_{i = 0}^{n} - 3 \cdot {(1.2)}^{n} .$

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Activity 8.3.11.

Given the closed forms you found in Activity 8.3.10, which of the following limits are defined? If defined, what is the limit?

$lim_{n \to \infty} A_{n} .$
$lim_{n \to \infty} B_{n} .$
$lim_{n \to \infty} C_{n} .$

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Definition 8.3.12.

Given a sequence $a_{n},$ we define the limit of the series

\sum_{n = k}^{\infty} a_{n} := lim_{n \to \infty} A_{n}

where $A_{n} = \sum_{i = k}^{n} a_{i} .$ We call $\sum_{n = k}^{\infty} a_{n}$ an infinite series.

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Activity 8.3.13.

Which of the following series are infinite?

$\sum_{n = 0}^{\infty} 3 (0.8)^{n} .$
$\sum_{n = 0}^{\infty} 2 {(\frac{5}{4})}^{n} .$
$\sum_{n = 0}^{\infty} {(\frac{5}{6})}^{n} .$
$\sum_{n = 0}^{\infty} \frac{1}{2} {(81)}^{n} .$
$\sum_{n = 0}^{\infty} 10 {(- \frac{1}{5})}^{n} .$

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Activity 8.3.14.

Let ${a_{n}}_{n = 1}^{\infty} = {\frac{1}{n} - \frac{1}{n + 1}} = 1 - \frac{1}{2}, \frac{1}{2} - \frac{1}{3}, \frac{1}{3} - \frac{1}{4}, \dots .$ Let $A_{n} = \sum_{i = 1}^{n} a_{i} = \sum_{i = 1}^{n} (\frac{1}{i} - \frac{1}{i + 1}) .$

Which of the following is the best strategy for evaluating $A_{4} = (1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + (\frac{1}{4} - \frac{1}{5}) ?$

Compute $A_{4} = (1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + (\frac{1}{4} - \frac{1}{5}) = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20},$ then evaluate the sum.
Rewrite $A_{4} = (1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + (\frac{1}{4} - \frac{1}{5}) = 1 + (- \frac{1}{2} + \frac{1}{2}) + (- \frac{1}{3} + \frac{1}{3}) + (- \frac{1}{4} + \frac{1}{4}) - \frac{1}{5},$ then simplify.

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Activity 8.3.15.

Recall from Activity 8.3.14 that ${a_{n}}_{n = 1}^{\infty} = {\frac{1}{n} - \frac{1}{n + 1}}$ and $A_{n} = \sum_{i = 1}^{n} a_{i} = \sum_{i = 1}^{n} (\frac{1}{i} - \frac{1}{i + 1}) .$

Compute the following partial sums:

$A_{3} .$
$A_{10} .$
$A_{100} .$

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Activity 8.3.16.

Recall from Activity 8.3.14 that ${a_{n}}_{n = 1}^{\infty} = {\frac{1}{n} - \frac{1}{n + 1}}$ and $A_{n} = \sum_{i = 1}^{n} a_{i} = \sum_{i = 1}^{n} (\frac{1}{i} - \frac{1}{i + 1}) .$

Which of the following is equal to $A_{n} ?$

$n - \frac{1}{n + 1} .$
$1 - \frac{1}{n} .$
$1 - \frac{1}{n + 1} .$
$1 - \frac{1}{i} .$
$1 - \frac{1}{i + 1} .$

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Definition 8.3.17.

Given a sequence ${x_{n}}_{1}^{\infty}$ and a sequence of the form ${s_{n}}_{1}^{\infty} := {x_{n} - x_{n + 1}}_{1}^{\infty}$ we call the series $S_{n} = \sum_{i = 1}^{n} s_{i} = \sum_{i = 1}^{n} (x_{i} - x_{i + 1})$ to be a telescoping series.

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Activity 8.3.18.

Given a telescoping series $S_{n} = \sum_{i = 1}^{n} s_{i} = \sum_{i = 1}^{n} (x_{i} - x_{i + 1}),$ find:

$S_{2} .$
$S_{10} .$
Choose $S_{n}$ from the following options:
1. $x_{1} - x_{n}$
2. $x_{1} - x_{n + 1}$
3. $x_{1} - x_{n - 1}$
4. $x_{1} - x_{n} + 1$
5. $x_{1} - x_{n} - 1$

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Activity 8.3.19.

For each of the following telescoping series, find the closed form for the $n$ th partial sum.

$S_{n} = \sum_{i = 1}^{n} (2^{- i} - (2^{- i - 1})) .$
$S_{n} = \sum_{i = 1}^{n} (i^{2} - (i + 1)^{2}) .$
$S_{n} = \sum_{i = 1}^{n} (\frac{1}{2 i + 1} - \frac{1}{2 i + 3}) .$

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Activity 8.3.20.

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?

$\sum_{i = 1}^{\infty} (2^{- i} - (2^{- i - 1})) .$
$\sum_{i = 1}^{\infty} (i^{2} - (i + 1)^{2}) .$
$\sum_{i = 1}^{\infty} (\frac{1}{2 i + 1} - \frac{1}{2 i + 3}) .$

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Activity 8.3.21.

Consider the partial sum sequence $A_{n} = (- 2) + (\frac{2}{3}) + (- \frac{2}{9}) + \dots + (- 2 \cdot {(- \frac{1}{3})}^{n}) .$

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(a)

A_{n}

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(b)

{A_{n}}

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Activity 8.3.22.

Consider the partial sum sequence $B_{n} = \sum_{i = 1}^{n} (\frac{1}{5 i + 2} - \frac{1}{5 i + 7}) .$

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(a)

B_{n}

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(b)

{B_{n}}

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Subsection 8.3.1 Videos

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Figure 151. 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.