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Section 8.3 Partial Sum Sequence (SQ3)

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 of \(\{a_n\}\) to be

\begin{equation*} A_k=\sum_{i=0}^k a_i=a_1+a_2+\cdots+a_k. \end{equation*}

Note that \(\{A_n\}=A_0, A_1, A_2, \ldots\) is itself a sequence called the partial sum sequence.

Activity 8.3.4.

Let \(a_n=\frac{2}{3^n}.\) Find the following partial sums:

  1. \(A_0\text{.}\)

  2. \(A_1\text{.}\)

  3. \(A_2\text{.}\)

  4. \(A_3\text{.}\)

  5. \(A_{100}\text{.}\)

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 \(a_n\text{,}\) we define the limit of the series

\begin{equation*} \displaystyle\sum_{n=k}^\infty a_n:=\lim_{n\to \infty} A_n \end{equation*}

where \(A_n=\displaystyle \sum_{i=k}^n a_i\text{.}\) We call \(\displaystyle\sum_{n=k}^\infty a_n\) an infinite series.

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)

\(A_n\)

(b)

\(\{A_n\}\)

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)

\(B_n\)

(b)

\(\{B_n\}\)

Subsection 8.3.1 Videos

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.