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

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Section 8.3 Partial Sums and Series (SQ3)

Subsection 8.3.1 Activities

Activity 8.3.1.

Consider the sequence \(\displaystyle \{a_n\}_{n=0}^\infty=\left\{\frac{1}{2^n}\right\}_{n=0}^\infty\text{.}\)
(a)
Find the first 5 terms of this sequence.
(b)
Compute the following:
  1. \(\displaystyle a_0.\)
  2. \(\displaystyle a_0+a_1.\)
  3. \(\displaystyle a_0+a_1+a_2.\)
  4. \(\displaystyle a_0+a_1+a_2+a_3.\)
  5. \(\displaystyle a_0+a_1+a_2+a_3+a_4.\)

Activity 8.3.2.

Consider the sequence \(\displaystyle \{a_n\}_{n=1}^\infty=\left\{\frac{1}{n}\right\}_{n=1}^\infty\text{.}\)
(a)
Find the first 5 terms of this sequence.
(b)
Compute the following:
  1. \(\displaystyle a_1.\)
  2. \(\displaystyle a_1+a_2.\)
  3. \(\displaystyle a_1+a_2+a_3.\)
  4. \(\displaystyle a_1+a_2+a_3+a_4.\)
  5. \(\displaystyle a_1+a_2+a_3+a_4+a_5.\)

Definition 8.3.3.

Given a sequence \(\{a_n\}_{n=0}^\infty\) define the \(k^{\text{th}}\) partial sum for this sequence to be
\begin{equation*} A_k=\sum_{i=0}^k a_i=a_0+a_1+a_2+\cdots+a_k. \end{equation*}
More generally, partial sums may be defined for any starting index. Given \(\{a_n\}_{n=N}^\infty\text{,}\) let
\begin{equation*} A_k=\sum_{i=N}^k a_i=a_N+a_{N+1}+a_{N+2}+\cdots+a_k. \end{equation*}

Activity 8.3.4.

Let \(a_n=\frac{2}{3^n}.\) Find the following partial sums of the sequence \(\{a_n\}_{n=0}^\infty\text{.}\)
(a)
\(A_0\text{.}\)
(b)
\(A_1\text{.}\)
(c)
\(A_2\text{.}\)
(d)
\(A_3\text{.}\)
(e)
\(A_{100}\text{.}\)

Activity 8.3.5.

Consider the sequence \(a_n=\frac{2}{3^n}.\) What is the best way to find the 100th partial sum \(A_{100}\text{?}\)
  1. Sum the first 101 terms of the sequence \(\{a_n\}\text{.}\)
  2. Find a closed form for the partial sum sequence \(\{A_n\}\text{.}\)

Activity 8.3.6.

Expand the following polynomial products, and then reduce to as few summands as possible.
  1. \((1-x)(1+x+x^2)\text{.}\)
  2. \((1-x)(1+x+x^2+x^3)\text{.}\)
  3. \((1-x)(1+x+x^2+x^3+x^4)\text{.}\)
  4. \((1-x)(1+x+x^2+\cdots+x^n)\text{,}\) where \(n\) is any nonnegative integer.

Activity 8.3.7.

Suppose \(\displaystyle S_5=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}.\) Without actually computing this sum, which of the following is equal to \(\left(1-\frac{1}{2}\right)S_5\text{?}\)
  1. \(\displaystyle\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}-\frac{1}{64}\text{.}\)
  2. \(\displaystyle1-\frac{1}{64}\text{.}\)
  3. \(\displaystyle1-\frac{1}{2}-\frac{1}{4}-\frac{1}{8}-\frac{1}{16}-\frac{1}{32}\text{.}\)

Activity 8.3.8.

Recall from Activity 8.3.4 that \(\displaystyle A_{100}=2+\frac{2}{3}+\frac{2}{3^2}+\frac{2}{3^3}+\frac{2}{3^4}+\cdots+\frac{2}{3^{100}}=2\left(1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+\cdots+\frac{1}{3^{100}}\right).\)
(a)
Which of the following is equal to \(\displaystyle\left(1-\frac{1}{3}\right)A_{100}\text{?}\)
  1. \(\displaystyle1-\frac{1}{3^{101}}\text{.}\)
  2. \(\displaystyle1-\frac{1}{3^{100}}\text{.}\)
  3. \(\displaystyle2\left(1-\frac{1}{3^{101}}\right)\text{.}\)
  4. \(\displaystyle2\left(1-\frac{1}{3^{100}}\right)\text{.}\)
(b)
Based on your previous choice, write out an expression for \(A_{100}\text{.}\)

Activity 8.3.9.

Suppose that \(\displaystyle \{b_n\}_{n=0}^\infty=\{(-2)^n\}_{n=0}^\infty=\{1,-2,4,-8,\ldots\}\text{.}\) Let \(B_n=\displaystyle\sum_{i=0}^n b_i\) be the \(n\)th partial sum of \(\{b_n\}\text{.}\)
(a)
Find simple expressions for the following:
  1. \((1-(-2))B_{10}\text{.}\)
  2. \((1-(-2))B_{30}\text{.}\)
  3. \((1-(-2))B_{n}\text{.}\) Choose from the following:
    1. \(1+(-2)^n\text{.}\)
    2. \(1-(-2)^n\text{.}\)
    3. \(1+(-2)^{n+1}\text{.}\)
    4. \(1-(-2)^{n+1}\text{.}\)
    5. \(1-2^n\text{.}\)
(b)
Based on your previous answers, solve for the following:
  1. \(B_{10}\text{.}\)
  2. \(B_{30}\text{.}\)
  3. \(B_{n}\text{.}\) Choose from the following:
    1. \(\displaystyle \displaystyle \frac{1-(-2)^{n+1}}{1-(-2)}\)
    2. \(\displaystyle \displaystyle \frac{1-(-2)^{n+1}}{1-2}\)
    3. \(\displaystyle \displaystyle \frac{1-(-2)^{n+1}}{1+(-2)}\)
    4. \(\displaystyle \displaystyle \frac{1-(-2)^{n}}{1-2}\)
    5. \(\displaystyle \displaystyle \frac{1-(-2)^{n}}{1-(-2)}\)

Activity 8.3.10.

Consider the following sequences:
  1. \(\displaystyle\{a_n\}_{n=0}^\infty=\left\{\left(-\frac{2}{3}\right)^n\right\}_{n=0}^\infty\text{.}\)
  2. \(\displaystyle\{b_n\}_{n=0}^\infty=\left\{2\cdot\left(-1\right)^n\right\}_{n=0}^\infty\text{.}\)
  3. \(\displaystyle\{c_n\}_{n=0}^\infty=\left\{-3\cdot \left(1.2\right)^n\right\}_{n=0}^\infty\text{.}\)
(a)
Find the closed form for the \(n\)th partial sum for the geometric sequence \(A_n=\displaystyle\sum_{i=0}^n a_i=\displaystyle\sum_{i=0}^n \left(-\frac{2}{3}\right)^n\text{.}\)
  1. \(\displaystyle \frac{3}{5}\left(1-\left(-\frac{2}{3}\right)^{n+1}\right)\text{.}\)
  2. \(\displaystyle \frac{5}{3}\left(1-\left(-\frac{2}{3}\right)^{n+1}\right)\text{.}\)
  3. \(\displaystyle \frac{5}{3}\left(1+\frac{2}{3}\left(\frac{2}{3}\right)^{n}\right)\text{.}\)
  4. \(\displaystyle \frac{3}{5}\left(1+\frac{2}{3}\left(\frac{2}{3}\right)^{n}\right)\text{.}\)
  5. \(\displaystyle 1-\left(-\frac{2}{3}\right)^{n+1}\text{.}\)
(b)
Find the closed form for the \(n\)th partial sum for the geometric sequence \(B_n=\displaystyle\sum_{i=0}^n b_i=\displaystyle\sum_{i=0}^n 2\cdot\left(-1\right)^n\text{.}\)
  1. \(\displaystyle 2^{n+1}\text{.}\)
  2. \(\displaystyle 1-(-1)^{n+1}\text{.}\)
  3. \(\displaystyle 1+(-1)^{n}\text{.}\)
  4. \(\displaystyle 2(1+(-1)^{n})\text{.}\)
  5. \(\displaystyle 2(1-(-1)^{n+1})\text{.}\)
(c)
Find the closed form for the \(n\)th partial sum for the geometric sequence \(C_n=\displaystyle\sum_{i=0}^n c_i=\displaystyle\sum_{i=0}^n -3\cdot \left(1.2\right)^n\text{.}\)

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?
  1. \(\displaystyle\lim_{n\to\infty}A_n\text{.}\)
  2. \(\displaystyle\lim_{n\to\infty}B_n\text{.}\)
  3. \(\displaystyle\lim_{n\to\infty}C_n\text{.}\)

Definition 8.3.12.

Given a sequence \(\{a_n\}_{n=k}^\infty\text{,}\) we define its infinite series (or just series) to be its sequence of its partial sums
\begin{equation*} \left\{A_n\right\}_{n=k}^\infty=\left\{\sum_{i=k}^n a_i\right\}_{n=k}^\infty =\left\{a_k,a_k+a_{k+1},a_k+a_{k+1}+a_{k+2},\dots\right\} \end{equation*}
and often use the notation
\begin{equation*} \sum_{i=k}^\infty a_i = a_k+a_{k+1}+a_{k+2}+\dots \end{equation*}
to represent it. We will also write \(\sum a_i\) for short when the starting index \(n=k\) is either known from context or irrelevant.
When the series (the sequence of partial sums) converges to a limit, we say the series is convergent and this limit is the value of the series, and write:
\begin{equation*} \sum_{i=k}^\infty a_i = a_k+a_{k+1}+a_{k+2}+\dots = \lim_{n\to\infty} \sum_{i=k}^n a_i\text{.} \end{equation*}
When the series (the sequence of partial sums) diverges, we say the series is divergent.

Activity 8.3.13.

Which of the following series are infinite?
  1. \(\displaystyle\sum_{n=0}^\infty 3(0.8)^n\text{.}\)
  2. \(\displaystyle\sum_{n=0}^\infty 2\left(\frac{5}{4}\right)^n\text{.}\)
  3. \(\displaystyle\sum_{n=0}^\infty \left(\frac{5}{6}\right)^n\text{.}\)
  4. \(\displaystyle\sum_{n=0}^\infty \frac{1}{2}\left(81\right)^n\text{.}\)
  5. \(\displaystyle\sum_{n=0}^\infty 10\left(-\frac{1}{5}\right)^n\text{.}\)

Activity 8.3.14.

Let \(\displaystyle\{a_n\}_{n=1}^\infty=\left\{\frac{1}{n}-\frac{1}{n+1}\right\}=1-\frac{1}{2}, \frac{1}{2}-\frac{1}{3}, \frac{1}{3}-\frac{1}{4},\ldots\text{.}\) Let \(\displaystyle A_n=\sum_{i=1}^na_i=\sum_{i=1}^n \left(\frac{1}{i}-\frac{1}{i+1} \right)\text{.}\)
Which of the following is the best strategy for evaluating \(\displaystyle A_{4}=\left(1-\frac{1}{2} \right)+\left(\frac{1}{2}-\frac{1}{3} \right)+\left(\frac{1}{3}-\frac{1}{4} \right)+\left(\frac{1}{4}-\frac{1}{5} \right)\text{?}\)
  1. Compute \(\displaystyle A_{4}=\left(1-\frac{1}{2} \right)+\left(\frac{1}{2}-\frac{1}{3} \right)+\left(\frac{1}{3}-\frac{1}{4} \right)+\left(\frac{1}{4}-\frac{1}{5} \right)=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}\text{,}\) then evaluate the sum.
  2. Rewrite \(\displaystyle A_{4}=\left(1-\frac{1}{2} \right)+\left(\frac{1}{2}-\frac{1}{3} \right)+\left(\frac{1}{3}-\frac{1}{4} \right)+\left(\frac{1}{4}-\frac{1}{5} \right)=1+\left(-\frac{1}{2}+\frac{1}{2}\right)+\left(-\frac{1}{3}+\frac{1}{3}\right)+\left(-\frac{1}{4}+\frac{1}{4}\right)-\frac{1}{5}\text{,}\) then simplify.

Activity 8.3.15.

Recall from Activity 8.3.14 that \(\displaystyle\{a_n\}_{n=1}^\infty=\left\{\frac{1}{n}-\frac{1}{n+1}\right\}\) and \(\displaystyle A_n=\sum_{i=1}^na_i=\sum_{i=1}^n \left(\frac{1}{i}-\frac{1}{i+1} \right)\text{.}\)
Compute the following partial sums:
  1. \(A_3\text{.}\)
  2. \(A_{10}\text{.}\)
  3. \(A_{100}\text{.}\)

Activity 8.3.16.

Recall from Activity 8.3.14 that \(\displaystyle\{a_n\}_{n=1}^\infty=\left\{\frac{1}{n}-\frac{1}{n+1}\right\}\) and \(\displaystyle A_n=\sum_{i=1}^na_i=\sum_{i=1}^n \left(\frac{1}{i}-\frac{1}{i+1} \right)\text{.}\)
Which of the following is equal to \(A_n\text{?}\)
  1. \(n-\frac{1}{n+1}\text{.}\)
  2. \(1-\frac{1}{n}\text{.}\)
  3. \(1-\frac{1}{n+1}\text{.}\)
  4. \(1-\frac{1}{i}\text{.}\)
  5. \(1-\frac{1}{i+1}\text{.}\)

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 \(S_n=\displaystyle\sum_{i=1}^n s_i=\sum_{i=1}^n(x_i-x_{i+1})\) to be a telescoping series.

Activity 8.3.18.

Given a telescoping series \(S_n=\displaystyle\sum_{i=1}^n s_i=\sum_{i=1}^n(x_i-x_{i+1})\text{,}\) find:
  1. \(S_2\text{.}\)
  2. \(S_{10}\text{.}\)
  3. Choose \(S_{n}\) from the following options:
    1. \(\displaystyle x_1-x_n\)
    2. \(\displaystyle x_1-x_{n+1}\)
    3. \(\displaystyle x_1-x_{n-1}\)
    4. \(\displaystyle x_1-x_n+1\)
    5. \(\displaystyle x_1-x_n-1\)

Activity 8.3.19.

For each of the following telescoping series, find the closed form for the \(n\)th partial sum.
  1. \(S_n=\displaystyle\sum_{i=1}^n (2^{-i}-(2^{-i-1}))\text{.}\)
  2. \(S_n=\displaystyle\sum_{i=1}^n (i^2-(i+1)^2)\text{.}\)
  3. \(S_n=\displaystyle\sum_{i=1}^n \left( \frac{1}{2i+1}-\frac{1}{2i+3}\right)\text{.}\)

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?
  1. \(\displaystyle\sum_{i=1}^\infty (2^{-i}-(2^{-i-1}))\text{.}\)
  2. \(\displaystyle\sum_{i=1}^\infty (i^2-(i+1)^2)\text{.}\)
  3. \(\displaystyle\sum_{i=1}^\infty \left( \frac{1}{2i+1}-\frac{1}{2i+3}\right)\text{.}\)

Activity 8.3.21.

Consider the partial sum sequence \(\displaystyle A_n=\left(-2\right)+\left(\frac{2}{3}\right)+\left(-\frac{2}{9}\right)+\cdots+\left(-2\cdot \left( -\frac{1}{3}\right)^n \right).\)
(a)
Find a closed form for \(A_n\text{.}\)
(b)
Does \(\{A_n\}\) converge? If so, to what value?

Activity 8.3.22.

Consider the partial sum sequence \(\displaystyle B_n=\sum_{i=1}^n \left( \frac{1}{5 \, i + 2}-\frac{1}{5 \, i + 7} \right).\)
(a)
Find a closed form for \(B_n\text{.}\)
(b)
Does \(\{B_n\}\) converge? If so, to what value?

Subsection 8.3.2 Videos

Figure 183. 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.

Subsection 8.3.3 Exercises

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