Series

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 11.2 – Series

Expected Educational Results

• Objective 19–01: I can explain the difference between a series and a sequence.

• Objective 19–02: I can classify a series as geometric, $p$$p$-series, harmonic, or telescoping.

• Objective 19–03: I can explain what it means for a series to converge and diverge.

• Objective 19–04: I can explain why ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0}$$\displaystyle\lim_{n\rightarrow\infty}{a_n}=0$ is a necessary but not sufficient condition for series convergence.

• Objective 19–05: I can use the appropriate Series Convergence Test to determine if a series converges or diverges.

• Objective 19–06: I can compute the interval on which a power series converges.

• Objective 19–07: I can compute the sum of a convergent geometric series.

• Objective 19–08: I can compute the sum of a telescoping series.

• Objective 19–09: I can use a geometric series to rewrite a repeating decimal as a rational number.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Series

Definition: Infinite Series (or Series)

The sum of all the terms of an infinite sequence, denoted

${\displaystyle \sum _{i=0}^{\mathrm{\infty }}{a}_i=a_0+a_1+a_2+\cdots +a_{i-1}+a_i+a_{i+1}+\cdots }$$\displaystyle\sum_{i=0}^{\infty}{a_i}=a_0+a_1+a_2+\cdots+a_{i-1}+a_i+a_{i+1}+\cdots$.

Partial Sums of a Series

Definition: Partial Sum of a Series, $S_n$$S_n$

${\displaystyle S_n=\sum _{i=0}^n{a}_i=a_0+a_1+\cdots +a_{n-1}+a_n}$$\displaystyle S_n =\sum_{i=0}^n{a_i}=a_0+a_1+\cdots +a_{n-1}+a_n$.

In other words,

$\begin{array}{rlr}{S}_0& =a_0& \\ S_1& =a_0+a_1& =S_0+a_1\\ S_2& =a_0+a_1+a_2& =S_1+a_2\\ S_3& =a_0+a_1+a_2+a_3& =S_2+a_3\end{array}$$\begin{align*}S_0 &= a_0 & \\ S_1 &= a_0 + a_1 &= S_0 + a_1 \\ S_2 &= a_0 + a_1 + a_2 &= S_1 + a_2 \\ S_3 &= a_0 + a_1 + a_2 + a_3 &= S_2 + a_3\end{align*}$

Investigation 03

Compute the first three partial sums, i.e., $S_0$$S_0$, $S_1$$S_1$, and $S_2$$S_2$ of the following series.

1. $a_n=\frac{n}{n^2+1}$$a_n=\frac{n}{n^2+1}$, $n\ge 1$$n\geq 1$

2. $a_n=\frac{1}{n+1}$$a_n=\frac{1}{n+1}$, $n\ge 0$$n\geq 0$

3. $a_n=(-1{)}^n\frac{1}{n^3+1}$$a_n=(-1)^n\frac{1}{n^3+1}$, $n\ge 0$$n\geq 0$

Check Your Work

Use Technology to Verify Partial Sums

DESMOS

Mathematica

NOTE: The use of technology in this activity is to compute partial sums. On Assessments, you may use technology to compute partial sums without showing work.

​x
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(* Investigation 03-1 *)
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(* Find first three partial sums of 1/(n+1), N>=0 *)
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(* First partial sum *)
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Sum[1/(n+1),{n,0,0}]
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(* Second partial sum *)
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Sum[1/(n+1),{n,0,1}]
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(* Third partial sum *)
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Sum[1/(n+1),{n,0,2}]
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(* For a decimal approximation of partial sum *)
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Sum[1/(n+1),{n,0,2}]//N

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. Sum[ ] has two arguments:

   1. The general $n^{\text{th}}$$n^{\text{th}}$-term, $a_n$$a_n$;

   2. $\{n,a,b\}$$\{n,a,b\}$, where

      1. $n$$n$ is the index (i.e., label for the terms);

      2. $a$$a$ is the initial label;

      3. $b$$b$ is the upper label.

3. For help on using the Sum[ ] function:

   1. In Mathematica, execute the code: $\text{?Sum}$$\text{?Sum}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the Sum[ ] function – you will be able to copy-paste code.

4. You may need parens, $($$(\,$ and $)$$\,)$, to group multiple terms in the numerator and denominator or to group $-1$$-1$ when raised to $n$$n$.

5. Remember, correct Mathematica code will be all black except for variables.

6. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Convergent Series

Definition: Convergent Series

A series is convergent if the sequence of partial sums, $\{S_n\}$$\{S_n\}$, is convergent, then ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{a}_n=S}$$\displaystyle\sum_{n=0}^{\infty}{a_n}=S$, where $S$$S$ is the [convergent] sum of the series.

NOTE: Using partial sums to determine if a series is convergent is algebraically-intense and time-consuming. Thus, we will use generalized rules, called convergence tests, to determine convergence of series.

Definition: Divergent Series

A series is divergent if the sequence of partial sums, $\{S_n\}$$\{S_n\}$, is divergent.

Homework

At this time, you should be able to complete the following assignments:

• Section 11.2: # 11, 15.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Wednesday, 27 October 2021 - 06:14 (EDT)