Preview For Chapter 12

Infinite Sequences and Series

The study of (infinite) series is widely regarded as the most difficult topic in all of calculus.  The idea of "adding up" an infinite number of terms and the "sum of the series" being a finite number sometimes and not a finite number other times seems to boggle the imagination. 

Sequences seem to be manageable, since the terms of an (infinite) sequence either approach a number (the limit) sufficiently or don't approach a number sufficiently, as n goes to infinity.  Our experience with limits of functions extends well to sequences.

The convergence of series to a sum is defined in terms of the convergence of sequences to a limit.  Clarifying the dual use of the word "converge" in the previous sentence is key in your success in dealing with series.  To determine convergence of series, you must master the concepts sequences.

Here are two reference sheets to print for tests for the convergence of sequences and series.

The List of Sequence and Series Tests For Convergence     Series Tests Summary Sheet
 

 

Calculus 2     Lesson 12.1     Sequences 
The following topics are the most important. Typical exercises from Calculus, 6th edition, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective
1. Define sequence, limit of a sequence, convergence of a sequence.   P720#1
P711 The opening paragraphs of this section give the definition of sequence.  Read the section through the notation and memorize the notation.  Look at examples 1 and 2.
P713 The definitions of limit of a sequence and convergence of a sequence are given in box 1.  Look at figures 4 and 5.
2. Find the formula for the general term {an}.   P720#11,13
P711 Look at example 1 to try to match the problem to one of these "types".
3. Determine whether a sequence converges or diverges using:
3a. "connect the dots", i.e., limit of f(x) by Chapter 2 methods;   P721#19,31,43
P714 The theorem in box 3 and figure 6 give the sense of this method.
3b. "connect the dots", i.e., limit of f(x) by l'Hopital's rule;   P721#31,35
P715 Example 4 illustrates the use of l'Hospital's Rule in finding the limit of the sequence.
3c. geometric sequence,  {a rn};  P721#22
P717 The convergence of geometric sequences rule is given in box 9 on this page.  Example 10 goes through the rationale.
3d. p-sequence,  {1/np};   P721#29
P714 See Box 4. Use the method in Example 4 or the limit law for ratios.
3e. sandwich theorem;   P721#47
P7 Use the squeeze theorem for sequences.
3f. absolute value to zero theorem;   P721#21
P715 Use the squeeze theorem for sequences to prove the theorem in box 6.
3g. bounded monotonic theorem;   P721#67
P717 The definitions for monotonic (both increasing and decreasing) are found in box 10.  Know the definitions.
P718 The definitions for bounded (both above and below) are found in box 11.  Know the definitions.
P719 The theorem for bounded, monotonic sequences is found in box 12.  Know the result.
3h. graphing calculator for directly or recursively defined sequences.   P721#47-52
P716 Look at the margin note at the bottom of the page and follow the directions.
4. Determine whether a sequence is monotone increasing or monotone decreasing by either using inequalities or by connecting the dots and using the derivative .   P721#59,60
P717 Look at examples 10 and 11.
5. State and explain the completeness property.   Text
P718 The Completeness Axiom is given in the paragraph below box 11.  Know the result.
For a synopsis of these tests, see Sequence tests

Maple Assignment: See Maple Assignment

 

Calculus 2     Lesson 12.2     Series
The following topics are the most important. Typical exercises from Calculus, 6th edition, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.
1. Define "series converges". Text P724
P724 The definition is given in box 2 on this page.  Know this thoroughly.
2. Find the sum of two types of convergent series.
2a. partial fractions series (telescoping series) P730#8,35,37
P726 The definition is given in the margin on this page.   Know this thoroughly.  Look at Example 6.
2b. geometric series P730#3,11,15,17
P725 The definition is given in box 4 on this page.   Know this thoroughly.  Look at Examples 1, 2, 3, 4, and 5.
3 Apply divergence tests.
3a. harmonic series Text P727
P727 The definition is given in Example 7 on this page.   Know this thoroughly.  We will be using this a lot.
3b. nth term test P730#9,23,25,31
P728 The definition is given in box 7 on this page.   Know this thoroughly.  Look at Example 8.
4. Express a repeating decimal as a ratio of integers. P731#41,45
P726 Example 4 shows how to use the geometric series to do this.
5. Given the nth partial sum, sn, of a series, find the terms, an, of the series and the limit of sn. Conclude whether the series converges and, if so, what the sum of the series is.   P731#55,56
P728 Review the definitions on this page and then go to the end of the section on page 729 and look at the theorems in box 8.  Use the two results as well as the series discussed in this section to help you solve these problems.
6. Find the sum or difference of two or more series.   P730#28
P729 The theorems in box 8 give the rules.   Look at Example 9.
Maple Assignment:  See Maple Assignment

 

Calculus 2     Lesson 12.3    The Integral Test 
The following topics are the most important. Typical exercises from Calculus, 6th edition, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.
1. Apply positive series tests.
1a. integral test P739#1,11,16,30
P735 The definition is given in the box on the top of this page.  Know this thoroughly.  Be sure that you test all three hypotheses: continuous, positive and decreasing when you use this test.  If the function is decreasing after a value c, use the integral test from c (or some integer greater than c) on.  Look at Examples 1, 2 and 4.
1b. p-series test P739#3,8,12
P736 The definition is given in box 1 on this page.   Know this thoroughly.  Look at Example 3.  We will be using this a lot.
2. Estimate the sum of a positive decreasing convergent series for a given number of terms, then improve the estimate. P740#32
P737 The definition is given in box 2 on this page.   Look at Examples 5 and 6.
Maple Assignment: none

 

Calculus 2     Lesson 12.4    Comparison Tests
The following topics are the most important. Typical exercises from Calculus, 6th edition, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.
1. Apply two more positive series tests.
1a. comparison test P745#1,2,3,7,11
P741 The definition for this test is given in the box in the middle of the page.  Look at Examples 1 and 2.
1b. limit comparison test P745#10,21,23,25
P743 The definition for this test is given in the box at the bottom of the page.  Look at Examples 3 and 4.
2. Estimate the sum of a positive decreasing convergent series, then estimate the error. P745#33
P744 The steps for this estimate are given the bottom of the page.  Look at Example 5.
Maple Assignment: none

 

Calculus 2     Lesson 12.5    Alternating Series 
The following topics are the most important. Typical exercises from Calculus, 6th edition, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.
1. Apply alternating series test.  P749#1,3,7,11,15
P746 The definition for this test is given in the box at the top of the page.  Look at Examples 1, 2 and 3.
2. Estimate the sum of an alternating series, then estimate the error. P750#21
P748 The definition for this estimate is given in the box at the middle of the page.  Look at Example 4.
3. Approximate the sum of alternating convergent series to an indicated accuracy. P750#27
P748 The definition for this test is given in the box at the middle of the page.  Look at Example 4.  Read the material on page 749.
Maple Assignment:  none

 

Calculus 2     Lesson 12.6    Absolute Convergence
The following topics are the most important. Typical exercises from Calculus, 6th edition, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.
1. Define "absolute convergence" and "conditional convergence". Text P750&751
P750 The definition of absolute convergence is given in box 1 on this page.  The theorem in box 3 on the next page is a result that you need to know.  Look at Example 1.
P751 The definition of conditional convergence is given in box 2 on this page.  Look at Example 2.
P751 To conclude that a series is conditionally convergent, you must use two tests: one to show that the series is not absolutely convergent and another to show that the series is convergent.
2. Determine whether a series is absolutely convergent, conditionally convergent or divergent using among other things,
2a. the previous methods, P755#5,10,17
P751 We use absolute convergence when the the terms do not alternate but some terms are positive and some are negative.  Look at Example 3.
2b. the ratio test, P755#1,3,8,13,31
P752 The definition is given in the box at the top of this page.  Look at Examples 4 and 5.
2c. the root test. P756#23,24
P754 The definition is given in the box in the middle of this page.  Look at Example 6.
Maple Assignment:  none

 

Calculus 2   Lesson 12.7   Strategy for Testing Series 
The following topics are the most important. Typical exercises from Calculus, 6th edition, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.
1. Use the strategy to decide which series convergence test to conduct and use that test to determine the convergence or divergence of a series. P#758#1,3,5,7,11,13,17,21,27
You may wish to refer to Dr.VanVelsir's Infinite Series Analysis Tree to see some of the strategies in flow chart form.
P757 The steps to use are given on this page.   Carefully examine Examples 1, 2, 3, 4, 5, and 6. Review the types of problems you had in the previous sections and do enough to get a sense of which type to try first.
For a synopsis of the series tests, click Series Tests

Use the series tree to show a flow chart for the process.

Maple Assignment:  none

Preview For Chapter 12-B

Power Series

As useful as sequences and series are, the real "power" in series comes when we introduce a variable  x  into the series.  Up to this point in time, the only variable in our series has been the index  n  running from 1 to infinity.  In power series, we keep the index  n  as before and introduce powers of  x,  i.e., xn, into the nth term of the series, hence the name power series.

Many popular functions like trigonometric functions and rational functions have power series representations.  A polynomial and its power series representation are one and the same.  Some functions like the Bessel functions are defined in terms of its power series representations.

The famous Maclaurin series and Taylor series, special cases of power series, have wide applications in physics and engineering.

 

Calculus 2     Lesson 12.8     Power Series
The following topics are the most important. Typical exercises from Calculus, 6th edition, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.
1. Determine the radius of convergence and interval of convergence for a power series. P763#3,7,11,20,21
P759 The definition of power series is given at box 1 on this page.
P759 The definition of power series centered at a is given at box 2 on this page.
P761 The definition of radius of convergence is given below box 3 on this page.
P761 The definition of interval of convergence is given below box 3 on this page.
P761 The theorem governing power series convergence is given in box 3 on this page.  The box at the bottom of the page gives the results of four examples.  See the note on page 760 regarding usually using the Ratio or Root Test and always checking both endpoints.  Look at Examples 1, 2, 3, 4, and 5.
Maple Assignment: See Maple Assignment

 

Calculus 2     Lesson 12.9     Representation of Functions as Power Series
The following topics are the most important. Typical exercises from Calculus, 6th edition, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.
1. Form new power series from existing power series using the following operations on power series and tell the radius or convergence of the new series.
1a. replace (x-a) by b( cx d)k in a series p769#3,4,6
P764 The definition of the sum of a geometric series is given in box 1 on this page.  From the basic formula, a/(1-u) = a + au + au 2 + ... decide what a and u are in the process.  Look at Example 1 if there is a 1 + u in the denominator.  Look at Example 2 if there is a constant b, not equal to 1, in the b + u in the denominator.
1b. multiply or divide a series by b(x-a)k p769#17,23
P765 Look at Example 3 if there is a variable term in the numerator.
1c. add or subtract two series p769#11
P764 After using partial fractions, use this definition in box 1 on this page for each of the fractions and combine as needed.
1d. differentiate a series p769#13
P766 The theorem governing differentiating the power series term by term is given in box 2 on this page.  Note that this process is only valid on the interval of convergence of the power series.  Look at Examples 4 and 5.
1e. integrate a series p769#20,27,29
P766 The theorem governing integrating the power series term by term is given in box 2 on this page.  Note that this process is only valid on the interval of convergence of the power series.  Look at Examples 6 and 7.
P768 Write the integrand as a power series, integrate term by term if the limits are within the interval of convergence, and then evaluate (or estimate) the result with the limits of integration.  Look at Example 8.
Maple Assignment:  none

 

Calculus 2    Lesson 12.10    Taylor and Maclaurin Series
The following topics are the most important. Typical exercises from Calculus, 6th edition, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.
1. Create the Taylor or Maclaurin series for a given function. P782#15,17,39
P771 The definition of a Taylor Series is found in box 6 on this page.
P772 The definition of a Maclaurin Series is found in box 7 on this page.
P775 The definition of the Maclaurin Series for ex  is found in box 11 on this page.
P779 The definitions of several Maclaurin Series are found in the box at the bottom of  this page.
P772 Examples 1 and 4 show the creation of a Maclaurin Series.
P775 Example 3 and 7 show the creation of a Taylor   Series.
2. From the Taylor or Maclaurin series for a function, form the series of a related function using the operations of section 12.9 above. P783#47,49,51,53
P776 Examples 5 and 6 show the creation of Maclaurin and Taylor Series.
3. Approximate values of the function using the Taylor or Maclaurin polynomial. P782#43
P777 Example 8 shows the evaluation of a Maclaurin Series.
4. Find the power series for a function that can be written as a binomial, f(x) = (1x)k , where k is any real number and 

|x| < 1.  P782#25-28
P778 Know the definition in box 17
P777 Read examples 8 and 9
Maple Assignment:  See Maple Assignment

 

Calculus 2     Lesson 12.11     Applications of Taylor Series
The following topics are the most important. Typical exercises from Calculus, 6th edition, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.
1. Find the Taylor or Maclaurin polynomial Tn(x) for a function f at a number a.
Show how well the Tn's approximate f by doing the following.
1a. Graph f and several Tn's on the same set of axes. P791#1,3,5
P785 The definition of Taylor polynomial Tn is given in the middle of the page.
P785 The definition of Taylor remainder Rn is given in the bottom of the page.
P785 The example in the left hand margin on this page gives the process.
1b. Evaluate f and several Tn's at the same x value. P791#23
P786 Example 1 shows this method on this page.
2. For a given function f,
2a. Approximate values of the function using the Taylor or Maclaurin polynomial.
P787 Example 2 part a shows this approach.
2b. Find an upper bound for the error above by bounding the remainder term using Taylor's inequality. P791#13,15
P787 Example 2 part b shows this approach.
Maple Assignment:  none