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MA102: Single-Variable Calculus II

Unit 5: Infinite Sequences and Series   *In this unit, you will become acquainted with the infinite lists called sequences and infinite sums called series.  The main question for each is whether it converges: do the terms of the sequence have a finite limit?  Do the series terms have a finite sum?  You will learn ways to test for convergence or divergence.  After learning a number of such tests, we will look at Taylor series, which are infinite polynomials.  Any function that may be differentiated an unlimited number of times gives rise to a Taylor series, whose partial sums are approximations to the function using ever higher-order derivatives.  We will consider questions like: for which values of the variable does the series converge?  For those values, is it equal to the function from which it was defined?

Many students find series the most difficult of the topics in Calculus II.  There are multiple expositions of each topic included in this unit, so be patient with yourself and study each resource carefully.*

Unit 5 Time Advisory
This unit should take you 29.5 hours to complete.

☐    Subunit 5.1: 4 hours

☐    Reading: 1 hour

☐    Lecture: 1 hour

☐    Assessment: 2 hours

☐    Subunit 5.2: 3.75 hours

☐    Subunit 5.3: 8.75 hours

☐    Sub-subunit 5.3.1: 1.75 hours

☐    Sub-subunit 5.3.2: 2 hours

☐    Sub-subunit 5.3.3: 1 hour

☐    Sub-subunit 5.3.4: 1.5 hours

☐    Sub-subunit 5.3.5: 1.5 hours

☐    Sub-subunit 5.3.6: 1 hour

☐    Subunit 5.4: 5 hours

☐    Sub-subunit 5.4.1: 1.5 hours

☐    Sub-subunit 5.4.2-5.4.4: 0.5 hours

☐    Sub-subunit 5.4.5: 0.5 hours

☐    Sub-subunit 5.4.6: 2.5 hours

☐    Subunit 5.5: 11.75 hours

☐    Sub-subunit 5.5.1: 4.25 hours

☐    Sub-subunit 5.5.2: 0.75 hours

☐    Sub-subunit 5.5.3: 4 hours

☐    Sub-subunit 5.5.4: 2.75 hours

Unit5 Learning Outcomes
Upon successful completion of this unit, the student will be able to: - Define convergence and limits in the context of sequences. - Define convergence and limits in the context of series. - Find the limits of sequences and series. - Discuss the convergence of the geometric and binomial series. - Show the convergence of positive series using the comparison, integral, limit comparison, ratio, and root tests. - Show the divergence of a positive series using the divergence test. - Show the convergence of alternating series. - Define absolute and conditional convergence. - Show the absolute convergence of a series using the comparison, integral, limit comparison, ratio, and root tests. - Manipulate power series algebraically. - Differentiate and integrate power series. - Compute Taylor and MacLaurin series. - Approximate functions using power series.

5.1 Sequences   A sequence is merely a list of terms (usually numbers) that are arranged in a particular order.  In this subunit, we will look at a sequence of numbers ordered by some rule or function.

  • Reading: Whitman College: David Guichard’s Calculus: Chapter 11: Sequences and Series: “Section 11.1 Sequences” Link: Whitman College: David Guichard’s Calculus: Chapter 11: Sequences and Series: “Section 11.1: Sequences” (PDF)

    Instructions: Please click on the link above, and read the brief introduction to chapter 11 and Section 11.1 (pages 253 through 260).

    Studying this reading should take approximately 1 hour.

    Terms of Use: The linked material above has been reposted by the kind permission of David Guichard, and can be viewed in its original form here (PDF).  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder. 

  • Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Sequences I” and “Sequences II” Link: University of Houston: Selwyn Hollis’s “Video Calculus: Sequences I” (QuickTime) Lecture and “Sequences II” (QuickTime)

    Instructions: Please click on the link, scroll down to Video 47: “Sequences I,” and watch the entire video.  Next, scroll down to Video 48: “Sequences II,” and watch it from the beginning through the 7th slide (the end of the slide marked 7 of 12).  If you are interested, feel free to watch the rest of the video.

    In the first video, Dr. Hollis discusses sequences and limits, goes over several important limits, and explains growth rates and order comparisons.  In the second video, he gives a precise definition of limits, shows how to do an epsilon-N proof for limits of sequences, and discusses boundedness and monotonicity.  In the optional slides (8-12), he discusses recursively-defined sequences, fixed points, and cobweb plots.

    Viewing these lectures and note-taking should take approximately 1 hour.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “Limits of Sequences” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “Limits of Sequences” (HTML)

    Instructions: Click on the link above.  Then, click on the “Index” button.  Scroll down to “1. Sequences,” and click button 184 (Limits of Sequences).  Do problems 1-22.  If at any time a problem set seems too easy for you, feel free to move on.

    Completing this assessment should take approximately 2 hours.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

5.2 Series   A series is the sum of the terms in an infinite sequence.  You are likely familiar with series arranged in an arithmetic or geometric progression; this subunit will take a look at terms defined by more intricate functions.  You can also view a series as another type of sequence – a sequence of partial sums.  In the following readings, you will learn what it means for a series to converge and study some important types of series.

5.2.1 Series and Basic Convergence   - Reading: Whitman College: David Guichard’s Calculus: Chapter 11: Sequences and Series: “Section 11.2 Series” Link: Whitman College: David Guichard’s Calculus: Chapter 11: Sequences and Series: “Section 11.2: Series” (PDF)

 Instructions: Please click on the link above, scroll down and read
Section 11.2 in its entirety (pages 260 through 263).  This section
will introduce you to infinite series and make mention of the
geometric series, which will be discussed in more detail below.  It
also explains what it means for a series to converge.  

 Studying this reading should take approximately 15-20 minutes.  

 Terms of Use: The linked material above has been reposted by the
kind permission of David Guichard, and can be viewed in its original
form
[here](http://www.whitman.edu/mathematics/multivariable/multivariable_11_Sequences_and_Series.pdf)
(PDF).  Please note that this material is under copyright and cannot
be reproduced in any capacity without explicit permission from the
copyright holder. 
  • Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Series” Link: University of Houston: Selwyn Hollis’s “Video Calculus: Series” (QuickTime)

    Instructions: This lecture will cover sub-subunits 5.2.1-5.3.1.  Please click on the link, scroll down to Video 49: “Series,” and watch the entire video.  You are welcome to break it into parts as you go along.  Slides 1-4 correspond roughly to 5.2.1; slides 5-7 correspond roughly to 5.2.3; slides 8-11 are an exposition of 5.2.2; and slide 12 corresponds to 5.3.1.

    Viewing this lecture and note-taking should take approximately 30 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

5.2.2 Properties of Infinite Series   - Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II “Special Series” Link: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Special Series” (HTML)

 Instructions: Please click on the link above, and read the
information on this webpage.  You may skip the section on
telescoping series; you are not responsible for that material. 
However, pay attention to the beginning through the discussion
following Example 2, pick up again at the paragraph before Example
5, and read to the end.  The notion that any finite number of terms
has no effect on the convergence behavior of a series is important
and can save you a lot of effort.  

 Studying this reading should take approximately 30 minutes.  

 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.2.3 Focus on the Geometric Series   A series is geometric if every successive term is the product of the previous term with a fixed value called the ratio of the series.  For example, 3 + 6 + 12 + 24 + ... is a geometric series with ratio 2.  Geometric series are unusual in that not only can we easily determine convergence or divergence for them, but in the case of convergence, we can also find the sum of the series exactly.

  • Reading: MIT: Gilbert Strang’s Calculus: Chapter 10: Infinite Series: “Section 10.1: The Geometric Series” Link: MIT: Gilbert Strang’s Calculus: Chapter 10: Infinite Series: “Section 10.1: The Geometric Series” (PDF)

    Instructions: Read the several paragraphs at the beginning of chapter 10 (the discussion of the geometric series begins here) and section 10.1 (pages 366-372).

    Studying this reading should take approximately 1 hour.

    Terms of Use: The article above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 (HTML).  It is attributed to Professor Gilbert Strang (MIT) and the original version can be found here (PDF).

  • Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Russo’s “Geometric Series” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Russo’s “Geometric Series” (HTML)

    Instructions: Click on the link above.  Then, click on the “Index” button.  Scroll down to “1.  Integration,” and click button 101 (Geometric Series).  Do problems 2-10.  If at any time a problem set seems too easy for you, feel free to move on.

    Completing this assessment should take approximately 1 hour.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

5.2.4 Highlight: The Binomial Series   If you wish to expand (1 + x)2 or (1 + x)3, you may simply multiply them out.  But what are the coefficients of the fifteen terms in the expansion of (1 + x)14?  The binomial theorem provides an answer in terms of combinations, also known as binomial coefficients.  The binomial series takes this even a step further, allowing the expansion of expressions such as 1/(1 + x) and (1 + x)1/2 to infinite polynomials.

  • Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Sequences and Series: Binomial Series” Link: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Sequences and Series: Binomial Series” (HTML)

    Instructions: Please click the link above, and read this entire section.  Recall that combinations arise from the problem of counting subsets of a collection of objects: the number of ways to choose two of the four letters ABCD is the combination 4 choose 2, which is 6.  If you have not seen it before, the connection to powers of binomials may not be clear.  To expand (1 + x)2, you write it as two copies of 1 + x and multiply one term from the first copy by one term from the second copy, using every possible pairing exactly once.  Likewise, to expand (1 + x)4, you must multiply across the four copies of (1 + x), taking every possible quartet exactly once: all the 1’s, the first two 1’s and the last two x’s, the first two x’s and the last two 1’s, etc.  Combinations allow you to find the coefficients, because, for example, the coefficient of x2 in the expansion of (1 + x)4 is 6: the number of quartets that contain exactly two x’s, or in other words the number of ways to select the two copies of (1 + x) that provide the x’s.

    The combination k choose n is typically written as the fraction (k!)/(n!(k-n)!).  Professor Dawkins writes it the way he does so it will generalize to negative and non-integer values of k.

    Studying this reading should take approximately 30 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

5.3 Test for Convergence of Positive Series   You will first learn how to check when a series with only positive terms converges – i.e. the limit of its sequence of partial sums exists and is finite.  The theory begins here with positive series, because it is the simplest problem to consider; these tests often work by “squeezing” the partial sums between zero and some larger series which are known to converge.

It is important to notice many of the tests related to series are one-directional implications: for example, if the series terms do not limit to zero, you can conclude the series diverges.  However, if it lacks that property, then you need more information to conclude either convergence or divergence.  Be careful not to assume a lack of one conclusion implies the opposite conclusion.

5.3.1 Divergence Test   Convergence of a series is convergence of its sequence of partial sums.  That is, for the series to converge, the partial sums must settle down and overall get closer and closer to a fixed finite value.  In order for that to happen, the amount being added to each partial sum to produce the next one must gradually shrink away to nothing.  That is the idea of the divergence test, which applies to any series (not just those with all positive terms): if the limit of the terms of the series is not zero, the series diverges.

This is not an equivalence, however!  Many divergent series have terms that limit to zero.  The terms must shrink to zero rapidly to give convergence.  However, whether the terms shrink to zero at all is straightforward to check and may save you work making more complicated tests on a divergent series.

  • Reading: MIT: Gilbert Strang’s Calculus: Chapter 10: Infinite Series: “Section 10.2: Convergence Tests: Positive Series” Link: MIT: Gilbert Strang’s Calculus: Chapter 10: Infinite Series: “Section 10.2 Convergence Tests: Positive Series” (PDF)

    Instructions: This reading will cover sub-subunits 5.3.1 – 5.3.6.  Please click on the link above, and read Section 10.2 in its entirety (pages 374 through 379).

    The section presents a criterion for divergence, the integral test, the comparison test, and the ratio and root tests.  We will revisit all these tests later in a slightly different context and give a more thorough justification of the last two tests.

    Studying this reading should take approximately 45 minutes.

    Terms of Use: The article above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 (HTML).  It is attributed to Professor Gilbert Strang (MIT) and the original version can be found here (PDF).

  • Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Divergence Test” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Divergence Test” (HTML)

    Instructions: Click on the link above.  Then, click on the “Index” button.  Scroll down to “2. Series,” and click button 187 (The Divergence Test).  Do problems 1-10.  If at any time a problem set seems too easy for you, feel free to move on.

    Completing this assessment should take approximately 1 hour.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

5.3.2 Integral Test   There is an imprecise correspondence between sequences and functions as well as between series and integrals.  The integral test shows this correspondence; though this relationship is not perfect, it is close enough to be useful.

  • Lecture: YouTube: MIT: David Jerison’s “Lecture 37: Infinite Series and Convergence Tests” Link: YouTube: MIT: David Jerison’s “Lecture 37: Infinite Series and Convergence Tests” (YouTube)

    Also Available in:
    iTunes U

    Instructions: This lecture will cover sub-subunits 5.3.2 - 5.3.4.  Please watch Lecture 37 from time 17:35 minutes to the end.  Note that lecture notes are available in PDF; the link is on the same page as the lecture.  Professor Jerison will begin his discussion of infinite series with the series [Sum of 1 divided by (n squared)].  He introduces important terminology and then proves the convergence of the above series by comparison with the integral of the summand.  He extends this argument to state the integral test.  Finally, he goes over the limit comparison test for positive sequences.

    Viewing this lecture and pausing to take notes should take approximately 45 minutes.

    Terms of Use: The video above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML).  It is attributed to MIT and the original version can be found here (Flash or MP4).

  • Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: The Integral Test” Link: University of Houston: Selwyn Hollis’s “Video Calculus: The Integral Test” (QuickTime)

    Instructions: Please click on the link above, scroll down to Video 50: “The Integral Test,” and watch this entire lecture.  This short video restates the integral test in a more concise way and provides several other important applications of the integral test, such as proving the convergence of the p-series and estimating remainders of partial sums.

    Viewing this lecture should take approximately 15 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Integral Test” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Integral Test” (HTML)

    Instructions: Click on the link above.  Then, click on the “Index” button.  Scroll down to “2. Series,” and click button 188 (The Integral Test).  Do problems 1-10.  If at any time a problem set seems too easy for you, feel free to move on.

    Completing this assessment should take approximately 1 hour.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

5.3.3 Comparison Test   Series with no negative terms either have terms that get small enough or fast enough for the series to converge, or terms that remain too large and hence cause the series to diverge.  It would seem logical that if Series A converges, and the nth term of Series B is less than or equal to the nth term of Series A for all n (at least after a finite number of terms), then Series B should converge: if A’s terms are small enough, B’s should also be.  Likewise, a series with terms that are larger than the corresponding terms of a divergent series should diverge.  This is true and is known as the (direct) comparison test.  

  • Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Comparison Tests” Link: University of Houston: Selwyn Hollis’s “Video Calculus: Comparison Tests” (QuickTime)

    Instructions: This lecture will cover sub-subunits 5.3.3-5.3.6.  Please click on the link above, scroll down to Video 51: “Comparison Tests,” and watch the video in its entirety.  This video states, proves, and applies the comparison, limit comparison, ratio, and root tests for positive series.

    Viewing this lecture and pausing to take notes should take approximately 30 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “5.4 Infinite Series: The Comparison Test” Link: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “5.4 Infinite Series: The Comparison Test” (PDF)

    Instructions: Please click the link above, and do problems 1 (a, c, e, g), 2 (a, c, e, g), and 4.  When finished, click here for solutions (courtesy of the author’s blog).

    Completing this assessment should take approximately 30 minutes.

    Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0.  It is attributed to Dan Sloughter and the original version can be found here (HTML). Please respect the copyright and terms of use displayed on the solution guide above.

5.3.4 Ratio Test   The ratio and root tests are two ways of checking whether a series is “geometric in the limit” and thereby drawing conclusions about its behavior.  The geometric series with terms arn, a and r positive, has the properties that the ratio of the n+1st term to the nth term is always r, and the nth root of the nth term is always r times the nth root of a.  The limit of each of those values as n tends to infinity is r.  The ratio test and root test check the limits of those values as computed from other series.  Although we lose some precision – i.e., a geometric series with ratio 1 diverges, but a limit of 1 in the ratio or root test is inconclusive – these tests greatly increase the number of series for which we can determine convergence or divergence.  Typically, only one of these tests is algebraically feasible for a given series, but in the event both are, note that if one is inconclusive, the other will be as well.

  • Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Ratio Test” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Ratio Test” (HTML)

    Instructions: Click on the link above.  Then, click on the “Index” button.  Scroll down to “2. Series,” and click button 189 (The Ratio Test).  Do problems 4-17.  (See the navigation buttons below the problem.)  If at any time a problem set seems too easy for you, feel free to move on.

    Completing this assessment should take approximately 1 hour and 30 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

5.3.5 Root Test   - Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The nth Root Test” Module Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The nth Root Test” (HTML)

 Instructions: Click on the link above.  Then, click on the “Index”
button.  Scroll down to “2. Series,” and click button 190 (The nth
Root Test).  Do problems 3-17.  (See the navigation buttons below
the problem.)  If at any time a problem set seems too easy for you,
feel free to move on.  

 Completing this assessment should take approximately 1 hour and 30
minutes.  

 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.3.6 Limit Comparison Test   Just as the ratio and root tests check whether a series is “geometric in the limit,” the limit comparison test checks whether two series are “equal in the limit,” up to a non-zero constant multiple.  If so, we can conclude the series have the same convergence behavior.

  • Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: “The Limit Comparison Test” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Limit Comparison Test” (HTML)

    Instructions: Click on the link above.  Then, click on the “Index” button.  Scroll down to “2. Series,” and click button 191 (The Limit Comparison Test).  Do problems 1-12.  (See the navigation buttons below the problem.)  If at any time a problem set seems too easy for you, feel free to move on.

    Completing this assessment should take approximately 1 hour.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: Clinton Community College: Elizabeth Wood’s Supplemental Notes for Calculus II: “Ratio and Root Test for Series of Nonnegative Terms” and “Infinite Series” Link: Clinton Community College: Elizabeth Wood’s Supplemental Notes for Calculus II:  “Ratio and Root Test for Series of Nonnegative Terms” (PDF) and “Infinite Series” (PDF)

    Also Available in:

    HTML (“Ratio and Root Test for Series of Nonnegative Terms”)

    HTML (“Infinite Series”)
     
    Instructions: This assessment will cover subunits 5.2-5.3.  Please click on the first link above and work through each of the seven examples on the page.   Next, please click on the second link and work through each of the thirteen examples on the page.  As in any assessment, solve the problem on your own first.  Solutions are given beneath each example.

    Completing this assessment should take approximately 2 hours.
     
    Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML) and here (HTML).  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder. 

5.4 Tests for Absolute and Conditional Convergence   Thus far, we have worked with series with all non-negative terms.  When series with negative terms are allowed, the picture changes slightly.  If all terms are negative, of course, the series is simply -1 times a series of all positive terms and has the same convergence behavior as the positive series.  If the terms are mixed sign, however, the positive and negative terms cancel each other out to some degree.  For such series, we have essentially two options: the alternating series test, which requires the signs alternate, or to take the absolute value of each term and test that series for convergence.

5.4.1 Alternating Series Test   An alternating series is one in which the terms alternate between positive and negative.  You may view it as a sequence of partial sums that alternately increase and decrease.  The alternating series test says that if the magnitudes of the terms decrease to zero in the limit, then the series converges.  For example, if every increase or decrease of the partial sums is smaller than the previous, and they limit to zero, the partial sums themselves have a finite limit.  Alternating series that are easy to write down tend either to diverge by the divergence test or converge by the alternating series test, but be aware that it is easy to define an alternating series with terms limiting to zero (but not decreasing to zero) that diverges.  For example, 1 −1/2 + 2/3 −1/3 + 1/2 −1/4 + ….  This is the harmonic series in disguise, as you will see if you pair off each positive term with the negative term following it.

  • Reading: Whitman College: David Guichard’s Calculus: Chapter 11: Sequences and Series: “Section 11.4: Alternating Series” Link: Whitman College: David Guichard’s Calculus: Chapter 11: Sequences and Series: “Section 11.4: Alternating Series” (PDF)

    Instructions: Please click on the link above, locate Chapter 11, and read Section 11.4 in its entirety (pages 269 through 273).

    Studying this reading should take approximately 30 minutes.

    Terms of Use: The linked material above has been reposted by the kind permission of David Guichard, and can be viewed in its original form here (PDF).  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder. 

  • Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Alternating Series and Absolute Convergence” Link: University of Houston: Selwyn Hollis’s “Video Calculus: Alternating Series and Absolute Convergence” (QuickTime)

    Instructions: This lecture will cover the topics outlined in sub-subunits 5.4.1 and 5.4.2 as well as 5.4.4 and 5.4.5.  Please click on the link above, scroll down to Video 52: “Alternating Series and Absolute Convergence,” and watch the video lecture in its entirety.  This video explains alternating series, conditional and absolute convergence, and the ratio and root tests for absolute convergence.

    Viewing this lecture and pausing to take notes should take approximately 30 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Alternating Series” Link: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Alternating Series” (PDF)

    Also Availabe in:

    HTML

    Instructions: Please click on the link above, and work through each of the five examples on the page.  As in any assessment, solve the problem on your own first.  Solutions are given beneath each example.

    Completing this assessment should take approximately 30 minutes.

    Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML).  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.

5.4.2 Definition of Absolute and Conditional Convergence   As we have seen with the harmonic and alternating harmonic series, the cancellation effect of a mixture of positive and negative terms can be vital for the convergence of a series.  The alternating harmonic is called a conditionally convergent series: its convergence is conditional on the cancellation effect.  If it is possible to eliminate the cancellation (by taking the absolute value of each term) and still have convergence, the series is called absolutely convergent.  When series have a mixture of positive and negative terms but do not alternate sign, taking the absolute value of the terms and testing the resulting series for convergence is often a good method.  It cannot tell you if the original series diverges, but it can show if the original series converges, absolutely.

  • Reading: MIT: Gilbert Strang’s Calculus: Chapter 10: Infinite Series: “Section 10.3: Convergence Tests: All Series” Link: MIT: Gilbert Strang’s Calculus: Chapter 10: Infinite Series: “Section 10.3: Convergence Tests: All Series” (PDF)

    Instructions: This reading will cover sub-subunits 5.4.2 and 5.4.3.  Please click on the link above, and read Section 10.3 in its entirety (pages 381 through 384).  It is worth pointing out that if a series converges conditionally, it does not have a well-defined sum.  By rearranging the terms, the value of the sum can be changed.  This is an example of the fact that infinity is a strange place.

    Studying this reading should take approximately 30 minutes.

    Terms of Use: The article above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 (HTML).  It is attributed to Professor Gilbert Strang (MIT) and the original version can be found here (PDF).

5.4.3 Comparison Test for Absolute Convergence   Note: This topic is covered by the reading assigned below sub-subunit 5.4.1.

5.4.4 Limit Comparison Test for Absolute Convergence   Note: This topic is covered by the lecture assigned below sub-subunit 5.4.1.

5.4.5 Ratio Test for Absolute Convergence   - Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Sequences and Series: Ratio Test” Link: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Sequences and Series: Ratio Test” (HTML)

 Instructions: Please click the link above, and read this entire
section.  This reading defines, applies, and proves the ratio
test*.*  

 Studying this reading should take approximately 30 minutes.  

 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.4.6 Root Test for Absolute Convergence   - Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Sequences and Series: Root Test” Link: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Sequences and Series: Root Test” (HTML)

 Instructions: Please click the link above, and read this entire
section.  This reading defines, applies, and proves the root test.  

 Studying this reading should take approximately 30 minutes.  

 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Assessment: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “5.6 Infinite Series: Absolute Convergence” Link: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “5.6 Infinite Series: Absolute Convergence” (PDF)

    Instructions: This assessment will cover sub-subunits 5.4.1-5.4.6.  Please click the link above, and do problems 1 (a, c, e, g), 2 (a, c, e, g), and 3 (a, b, c).  When finished, click here for solutions (courtesy of the author’s blog).

    Completing this assessment should take approximately 45 minutes.

    Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0.  It is attributed to Dan Sloughter and the original version can be found here (HTML). Please respect the copyright and terms of use displayed on the solutions guide above.

  • Assessment: Millersville University: Bruce Ikenaga’s “Absolute Convergence and Conditional Convergence” Link: Millersville University: Bruce Ikenaga’s “Absolute Convergence and Conditional Convergence” (HTML)

    Instructions: This assessment will cover sub-subunits 5.4.1-5.4.6.  Please click on the link above and scroll down the page.  Work through the last four examples before looking at their solutions.

    Completing this assessment should take approximately 30 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Absolute and Conditional Convergence” Link: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Absolute and Conditional Convergence” (PDF)

    Also Available in:

    HTML

    Instructions: This assessment will cover sub-subunits 5.4.1-5.4.6.  Please click on the link above, and work through each of the six examples on the page.  As in any assessment, solve the problem on your own first.  Detailed solutions are given beneath each example.

    Completing this assessment should take approximately 45 minutes.

    Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML).  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.

5.5 Series Representations of Functions   In Single-Variable Calculus I (MA101), we learned that we can approximate a function about a point when we have information about the function’s value and its slope at that point.  In this subunit, you will learn how to be even more accurate by gathering additional information about the function at a particular point (i.e. by using the second derivative, third derivative, fourth derivative, etc.).  The more information you collect, the closer you will get to the function itself.  The series representation of a function is the infinite series about a point, taking into consideration all of the derivatives about that point and in the form of a polynomial.  This will enable us to look at functions, derivatives, and integrals in new and rather intuitive ways.

5.5.1 Power Series   A power series is any series where the nth term contains xn (where x as the name of the variable and the exact matching of the term index with the exponent are not essential).  Essentially, it is an infinite polynomial in x.

  • Reading: Whitman College: David Guichard’s Calculus: “Chapter 11: Sequences and Series: Section 11.8: Power Series” Link: Whitman College: David Guichard’s Calculus: “Chapter 11: Sequences and Series: Section 11.8: Power Series” (PDF)

    Instructions: Please click on the link above and read Section 11.8 in its entirety (pages 278 through 281).  A key term to understand in this section is radius of convergence.

    Studying this reading should take approximately 15-20 minutes.

    Terms of Use: The linked material above has been reposted by the kind permission of David Guichard, and can be viewed in its original form here (PDF).  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder. 

  • Lecture: YouTube: MIT: David Jerison’s “Lecture 38: Taylor Series” and Haynes Miller’s “Lecture 39: Final Review” Link: YouTube: MIT: David Jerison’s “Lecture 38: Taylor Series” (YouTube)

    Also Available in:
    iTunes U

    and Haynes Miller’s “Lecture 39: Final Review (YouTube)

    Also Available in:
    iTunes U

    Instructions: These lectures cover subunits 5.5.1-5.5.3. After completing the readings for these sections, click on the first link, and watch Lecture 38 from the 22:45 minute mark to the end. In this lecture, Professor Jerison will discuss general power series before introducing Taylor Series. Then, watch Lecture 39.  Professor Miller will continue this exposition; he will go over the derivations for the power series for the exponential, the sine, and the cosine before moving on to other examples. 
    Note that on the original pages (linked below), the lecture notes are available in PDF under the "Related Resources" tab. 

    Watching these lectures and pausing to take notes should take approximately 1 hour and 15 minutes.

    Terms of Use: The videos above are released under a Creative Commons Attribution-Share-Alike License 3.0. They are attributed to MIT and the original versions can be found here and here.

  • Reading: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Power Series and the Uses of Power Series” Link: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Power Series and the Uses of Power Series” (PDF)

    Also Available in:

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    Instructions: Please click on the link above, and work through each of the six examples on the page.  As in any assessment, solve the problem on your own first.  Solutions are given beneath each example.

    Completing this assessment should take approximately 45 minutes.

    Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML).  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.

  • Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “Power Series” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “Power Series” (HTML)

    Instructions: Click on the link above.  Then, click on the “Index” button.  Scroll down to “3. Power and Taylor Series,” and click button 192 (Power Series).  Do the problems in the module as they are presented to you (18 total).  These problems all deal with computing the radius of convergence for a power series.  If at any time a problem set seems too easy for you, feel free to move on.

    Completing this assessment should take approximately 2 hours.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

5.5.2 Calculus with Power Series   Happily, our infinite polynomials interact with integration and differentiation in the same way as finite polynomials: the integral of a sum is the sum of the integrals of each term, and likewise for derivatives.  The bookkeeping aspects of this topic will be easier if you think of the “point of view” of the mathematical operators: the integral sign and derivative operator d/dx see x as a variable and n as a constant (a different fixed value for each series term).  On the other hand, the summation operator sees n as the variable and x as the constant (a value that will be the same for every series term).

  • Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 9: Infinite Series: “Section 9.8: Derivatives and Integrals of Power Series” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 9: Infinite Series: “Section 9.8: Derivatives and Integrals of Power Series” (PDF)

    Instructions: Please click on the link above, and read Section 9.8 in its entirety (pages 533 through 539).

    Under certain conditions, power series can be differentiated and integrated.  Certain characteristics of the series may change, however, such as the interval of convergence.  We have not been using Keisler’s text so far this unit, because hyperreals are not as helpful to the intuition for series as they are for integrals.  In the last several sub-subunits, they do not appear except in the proof of Theorem 1, part (iii), on pages 537-538 of this section.  That statement, that the radius of convergence remains the same when a power series is integrated or differentiated, is typically given in calculus texts without proof.  Do not fret too much over the proof.

    Studying this reading should take approximately 45 minutes to complete.

    Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML).  It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).

5.5.3 Taylor and Maclaurin Series   Taylor series are a particular kind of power series, defined from a function.  Maclaurin series are a particular kind of Taylor series.  The definition allows you to expand any function into an infinite polynomial, which in many cases will be provably equal to the original function, and may be much easier to compute with.

  • Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 9: Infinite Series: “Section 9.10: Taylor’s Theorem” and “Section 9.11: Taylor Series” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 9: Infinite Series: “Section 9.10: Taylor’s Theorem” (PDF) and “Section 9.11: Taylor Series” (PDF)

    Instructions: Please click on the links above, and read Sections 9.10 and 9.11 (pages 547 through 560).

    Taylor Series use the information provided by the derivatives of a function (slope of the tangent line, concavity, etc.) to approximate the function by a sequence of polynomials.  Taylor’s Theorem tells how good this approximation is.  Taylor Series are used extensively in higher mathematics, especially in numerical analysis.

    Studying these readings should take approximately 2 hours.

    Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML).  It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).

  • Assessment: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Taylor and MacLaurin Series” Link: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Taylor and MacLaurin Series” (PDF)

    Also Available in:

    HTML

    Instructions: Please click on the link above, and work through each of the seven examples on the page.  As in any assessment, solve the problem on your own first.  Solutions are given beneath each example.

    Completing this assessment should take approximately 1 hour.

    Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML).  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.

  • Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “Taylor Series” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “Taylor Series” (HTML)

    Instructions: Click on the link above.  Then, click on the “Index” button.  Scroll down to “3. Power and Taylor Series,” and click button 193 (Taylor Series).  Choose at least 5 of the problems to do.  If at any time a problem set seems too easy for you, feel free to move on.

    Completing this assessment should take approximately 1 hour.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

5.5.4 Approximation by Power Series   - Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 9: Infinite Series: “Section 9.9: Approximations by Power Series” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 9: Infinite Series: “Section 9.9: Approximations by Power Series” (PDF)

 Instructions: Please click on the link above, and read Section 9.9
in its entirety (pages 540 through 546).  Approximation by power
series is a very important topic; for instance, it is how
calculators compute sines and cosines!  

 Studying this reading should take approximately 1 hour.  

 Terms of Use: The article above is released under a [Creative
Commons Attribution-NonCommercial-ShareAlike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/) (HTML).  It
is attributed to H. Jerome Kiesler and the original version can be
found [here](http://www.math.wisc.edu/~keisler/calc.html) (PDF).
  • Web Media: UC College Prep’s Calculus BC II for AP: “Infinite Sequences and Series: Approximating Functions Using Polynomials” and “Infinite Sequences and Series: Applications of Taylor Series” Link:  UC College Prep’s Calculus BC II for AP: “Infinite Sequences and Series: Approximating Functions Using Polynomials” (YouTube) and “Infinite Sequences and Series: Applications of Taylor Series” (YouTube)

    Instructions: Click on the links above, and watch the interactive lectures.  You may want to have a pencil and paper close by, as you will be prompted to work on related problems during the lecture. 

    Studying these lectures should take approximately 1 hour.

    Terms of Use:  The videos above are licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives License.  They are attributed to University of California College Prep.

  • Assessment: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Other Topics Related to Taylor Series” Link: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Other Topics Related to Taylor Series” (PDF)

    Also Available in:

    HTML

    Instructions: Click on the link above, and work through each of the five examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

    Completing this assessment should take approximately 45 minutes.

    Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here.  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.