Tag Archives: Convergence

Unit 10 – Infinite Sequences and Series

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Unit 10 covers sequences and series. These are BC only topics (CED – 2019 p. 177 – 197). These topics account for about 17 – 18% of questions on the BC exam.

Topic 10.1: Defining Convergent and Divergent Series.

Topic 10. 2: Working with Geometric Series. Including the formula for the sum of a convergent geometric series.

Topics 10.3 – 10.9 Convergence Tests

The tests listed below are assessed on the BC Calculus exam. Other methods are not tested. However, teachers may include additional methods.

Topic 10.3: The nth Term Test for Divergence.

Topic 10.4: Integral Test for Convergence. See Good Question 14

Topic 10.5: Harmonic Series and p-Series. Harmonic series and alternating harmonic series, p-series.

Topic 10.6: Comparison Tests for Convergence. Comparison test and the Limit Comparison Test

Topic 10.7: Alternating Series Test for Convergence.

Topic 10.8: Ratio Test for Convergence.

Topic 10.9: Determining Absolute and Conditional Convergence. Absolute convergence implies conditional convergence.

Topics 10.10 – 10.12 Taylor Series and Error Bounds

Topic 10.10: Alternating Series Error Bound.

Topic 10.11: Finding Taylor Polynomial Approximations of a Function.

Topic 10.12: Lagrange Error Bound.

Topics 10.13 – 10.15 Power Series

Topic 10.13: Radius and Interval of Convergence of a Power Series. The Ratio Test is used almost exclusively to find the radius of convergence. Term-by-term differentiation and integration of a power series gives a series with the same center and radius of convergence. The interval may be different at the endpoints.

Topic 10.14: Finding the Taylor and Maclaurin Series of a Function. Students should memorize the Maclaurin series for \displaystyle \frac{1}{{1-x}}, sin(x), cos(x), and ex.

Topic 10.15: Representing Functions as Power Series. Finding the power series of a function by differentiation, integration, algebraic processes, substitution, or properties of geometric series.

Timing

The suggested time for Unit 9 is about 17 – 18 BC classes of 40 – 50-minutes, this includes time for testing etc.

Previous posts on these topics:

Before sequences

Amortization Using finite series to find your mortgage payment. (Suitable for pre-calculus as well as calculus)

A Lesson on Sequences.  An investigation, which could be used as early as Algebra 1, showing how irrational numbers are the limit of a sequence of approximations. Also, an introduction to the Completeness Axiom. 

Everyday Series

Convergence Tests

Reference Chart

Which Convergence Test Should I Use? Part 1: Pretty much anyone you want!

Which Convergence Test Should I Use? Part 2: Specific hints and a discussion of the usefulness of absolute convergence

Good Question 14 on the Integral Test

Sequences and Series

Graphing Taylor Polynomials.  Graphing calculator hints

Introducing Power Series 1

Introducing Power Series 2

Introducing Power Series 3

New Series from Old 1: Substitution (Be sure to look at example 3)

New Series from Old 2: Differentiation

New Series from Old 3: Series for rational functions using long division and geometric series

Geometric Series – Far Out: An instructive “mistake.”

A Curiosity: An unusual Maclaurin Series

Synthetic Summer Fun Synthetic division and calculus including finding the (finite)Taylor series of a polynomial.

Adapting 2021 BC 5

Adapting 2021 BC 6

Error Bounds

Error Bounds: Error bounds in general and the alternating Series error bound, and the Lagrange error bound

The Lagrange Highway: The Lagrange error bound. 

What’s the “Best” Error Bound?

Review Notes

Type 10: Sequences and Series Questions

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Adapting 2021 BC 6

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Nine of nine. We continue our look at the 2021 free-response questions. We will look at ways to adapt, expand, and explore this question to help students better understand it and look at other questions that can be asked based on a similar stem.

2021 BC 6

This is a Sequence and Series (Type 10) question. Typically the topic of the last question on the BC exam, it tests the concepts in Unit 10 of the current Calculus Course and Exam Description. This year the previous question, 2021 BC 5, asked students to write a Taylor Polynomial. This question covers other related topics: convergence tests, radius of convergence, and the error bound.

There is a nuance here. In past years students were not asked to give the conditions for a convergence test and were expected to determine which test to use for themselves. I think the idea here, and perhaps going forward (?), is to make sure the students have considered the conditions necessary to use a test. This is in keeping with other questions where the hypotheses of the theorem students were using had to be checked (Cf. recent L’Hospital’s Rule questions).

The Convergence Test Chart  and the posts “Which Convergence Test Should I use?” Part 1 and Part 2 may be helpful.

The stem for 2021 BC 6 is:

Part (a): Students were asked to give the conditions for the integral test and use it to determine if a different series, \sum\limits_{{n=0}}^{\infty }{{\frac{1}{{{{e}^{n}}}}}} converges.

Discussion and ideas for adapting this question:

  • Be sure your students know the conditions necessary for each convergence test. Phrase your questions as this one is phrased – at least sometimes.
  • Ask students to state the conditions for any convergence they use.
  • Discuss which tests (often plural) can be used for each series you study.
  • Make sure students can decide for themselves which test to use in case next year’s questions do not tell them.
  • Ask what other test(s) may be used with this series (Hint: the series is geometric). This is a question to ask for any series you study.

Part (b): Students are told to use the series from part (a) with the limit comparison test to show that the given series converges absolutely when x = 1. Again, students were asked to use a specific test. Notice that even if a student could not do part (a), they were not shut out of part (b).

Discussion and ideas for adapting this question:

  • Since you cannot count on being told which test to use for comparison, be sure to discuss how to decide which test(s) can be used with each series. Again see “Which Convergence Test Should I use?” Part 1 and Part 2.
  • Show students that proving absolute convergence is often a good way to eliminate the need for dealing with alternating series and other series with negative signs.

Part (c): Students were asked for the radius of convergence of the series. A standard question done by using the Ratio test.

Discussion and ideas for adapting this question:

The only extension here is to determine the interval of convergence, by checking the endpoints.

Part (d): Students were asked for the alternating series error bound using the first two terms to approximate the value of g(1). Even though there are only two error bounds students are expected to be able to compute (the other is the Lagrange error bound), students were again told which one to use. The result was not expected to be expressed as a decimal.

Discussion and ideas for adapting this question:

  • First, have students check that the conditions for using the alternating series error bound are met.
  • Increase the number of terms to be used.
  • Ask students to find the Lagrange error bound and compare the results.

This post on the series question concludes the series of posts (pun intended) considering how to expand and adapt the 2021 AP Calculus free-response questions. I hope you found them helpful.

As always, I happy to hear your ideas for other ways to use this question. Please share your thoughts and ideas.

Infinite Sequences and Series – Unit 10

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Unit 10 covers sequences and series. These are BC only topics (CED – 2019 p. 177 – 197). These topics account for about 17 – 18% of questions on the BC exam.

Topics 10.1 – 10.2

Topic 10.1: Defining Convergent and Divergent Series.

Topic 10. 2: Working with Geometric Series. Including the formula for the sum of a convergent geometric series.

Topics 10.3 – 10.9 Convergence Tests

The tests listed below are tested on the BC Calculus exam. Other methods are not tested. However, teachers may include additional methods.

Topic 10.3: The nth Term Test for Divergence.

Topic 10.4: Integral Test for Convergence. See Good Question 14

Topic 10.5: Harmonic Series and p-Series. Harmonic series and alternating harmonic series, p-series.

Topic 10.6: Comparison Tests for Convergence. Comparison test and the Limit Comparison Test

Topic 10.7: Alternating Series Test for Convergence.

Topic 10.8: Ratio Test for Convergence.

Topic 10.9: Determining Absolute and Conditional Convergence. Absolute convergence implies conditional convergence.

Topics 10.10 – 10.12 Taylor Series and Error Bounds

Topic 10.10: Alternating Series Error Bound.

Topic 10.11: Finding Taylor Polynomial Approximations of a Function.

Topic 10.12: Lagrange Error Bound.

Topics 10.13 – 10.15 Power Series

Topic 10.13: Radius and Interval of Convergence of a Power Series. The Ratio Test is used almost exclusively to find the radius of convergence. Term-by-term differentiation and integration of a power series gives a series with the same center and radius of convergence. The interval may be different at the endpoints.

Topic 10.14: Finding the Taylor and Maclaurin Series of a Function. Students should memorize the Maclaurin series for \displaystyle \frac{1}{{1-x}}, sin(x), cos(x), and ex.

Topic 10.15: Representing Functions as Power Series. Finding the power series of a function by, differentiation, integration, algebraic processes, substitution, or properties of geometric series.

Timing

The suggested time for Unit 9 is about 17 – 18 BC classes of 40 – 50-minutes, this includes time for testing etc.

Previous posts on these topics:

Before sequences

Amortization Using finite series to find your mortgage payment. (Suitable for pre-calculus as well as calculus)

A Lesson on Sequences An investigation, which could be used as early as Algebra 1, showing how irrational numbers are the limit of a sequence of approximations. Also, an introduction to the Completeness Axiom. 

Everyday Series

Convergence Tests

Reference Chart

Which Convergence Test Should I Use? Part 1 Pretty much anyone you want!

Which Convergence Test Should I Use? Part 2 Specific hints and a discussion of the usefulness of absolute convergence

Good Question 14 on the Integral Test

Sequences and Series

Graphing Taylor Polynomials Graphing calculator hints

Introducing Power Series 1

Introducing Power Series 2

Introducing Power Series 3

New Series from Old 1 substitution (Be sure to look at example 3)

New Series from Old 2 Differentiation

New Series from Old 3 Series for rational functions using long division and geometric series

Geometric Series – Far Out An instructive “mistake.”

A Curiosity An unusual Maclaurin Series

Synthetic Summer Fun Synthetic division and calculus including finding the (finite)Taylor series of a polynomial.

Error Bounds

Error Bounds Error bounds in general and the alternating Series error bound, and the Lagrange error bound

The Lagrange Highway The Lagrange error bound. 

What’s the “Best” Error Bound?

Review Notes

Type 10: Sequences and Series Questions

 

 

 

 

 

2019 CED Unit 10: Infinite Sequences and Series

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Unit 10 covers sequences and series. These are BC only topics (CED – 2019 p. 177 – 197). These topics account for about 17 – 18% of questions on the BC exam.

Topics 10.1 – 10.2

Timing

The suggested time for Unit 9 is about 17 – 18 BC classes of 40 – 50-minutes, this includes time for testing etc.

Previous posts on these topics :

Introducing Power Series 1

Which Convergence Test Should I Use? Part 1

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One common question from students first learning about series is how to know which convergence test to use with a given series.  The first answer is: practice, practice, practice. The second answer is that there is often more than one convergence test that can be used with a given series.

I will illustrate this point with a look at one series and the several tests that may be used to show it converges. This will serve as a review of some of the tests and how to use them. For a list of convergence tests that are required for the AP Calculus BC exam click here.

To be able to use these tests the students must know the hypotheses of each test and check that they are met for the series in question. On multiple-choice questions students do not need to how their work, but on free-response questions (such as checking the endpoints of the interval of convergence of a Taylor series) they should state them and say that the series meets them.

For our example we will look at the series \displaystyle 1-\frac{1}{3}+\frac{1}{9}-\frac{1}{{27}}+\frac{1}{{81}}-+\ldots =\sum\limits_{{n=1}}^{\infty }{{{{{\left( {-\frac{1}{3}} \right)}}^{{n-1}}}}}

Spoiler: Except for the first two tests to be considered, the other tests are far more work than is necessary for this series. The point is to show that several tests may be used for a given series, and to practice the other tests.

The Geometric Series Test is the obvious test to use here, since this is a geometric series. The common ratio is (–1/3) and since this is between –1 and 1 the series will converge.

The Alternating Series Test (the Leibniz Test) may be used as well. The series alternates signs, is decreasing in absolute value, and the limit of the nth term as n approaches infinity is 0, therefore the series converges.

The Ratio Test is used extensively with power series to find the radius of convergence, but it may be used to determine convergence as well. To use the test, we find

\displaystyle \underset{{n\to \infty }}{\mathop{{\lim }}}\,\frac{{\left| {{{{\left( {-\frac{1}{3}} \right)}}^{{n+1}}}} \right|}}{{\left| {{{{\left( {-\frac{1}{3}} \right)}}^{n}}} \right|}}=\frac{1}{3}  Since the limit is less than 1, we conclude the series converges.

Absolute Convergence

A series, \sum\limits_{{n=1}}^{\infty }{{{{a}_{n}}}}, is absolutely convergent if, and only if, the series \sum\limits_{{n=1}}^{\infty }{{\left| {{{a}_{n}}} \right|}} converges. In other words, if you make all the terms positive, and that series converges, then the original series also converges. If a series is absolutely convergent, then it is convergent. (A series that converges but is not absolutely convergent is said to be conditionally convergent.)

The advantage of going for absolute convergence is that we do not have to deal with the negative terms; this allows us to use other tests.

Applied to our example, if the series \sum\limits_{{n=1}}^{\infty }{{{{{\left( {\frac{1}{3}} \right)}}^{{n-1}}}}} converges, then our series \sum\limits_{{n=1}}^{\infty }{{{{{\left( {-\frac{1}{3}} \right)}}^{{n-1}}}}} will converge absolutely and converge.

The Geometric Series Test can be used again as above.

The Integral Test says if the improper integral \displaystyle {{\int_{1}^{\infty }{{\left( {\frac{1}{3}} \right)}}}^{x}}dx converges, then our original series will converge absolutely.

\displaystyle \int\limits_{1}^{\infty }{{{{{\left( {\frac{1}{3}} \right)}}^{x}}}}dx=\underset{{n\to \infty }}{\mathop{{\lim }}}\,\int\limits_{1}^{n}{{{{{\left( {\frac{1}{3}} \right)}}^{x}}}}dx=\underset{{n\to \infty }}{\mathop{{\lim }}}\,\left( {\frac{{{{{\left( {\frac{1}{3}} \right)}}^{n}}}}{{\ln \left( {1/3} \right)}}-\frac{{{{{\left( {\frac{1}{3}} \right)}}^{1}}}}{{\ln \left( {1/3} \right)}}} \right)=0-\frac{{1/3}}{{\ln \left( {1/3} \right)}}

\displaystyle =-\frac{{1/3}}{{\ln \left( {1/3} \right)}}>0 since ln(1/3) < 0.

The limit is finite, so our series converges absolutely, and therefore converges.

The Direct Comparison Test may also be used. We need to find a positive convergent series whose terms are term-by-term greater than the terms of our series. The geometric series \sum\limits_{{n=1}}^{\infty }{{{{{\left( {\frac{1}{2}} \right)}}^{{n-1}}}}} meets these two requirements. Therefore, the original series converges absolutely and converges.

The Limit Comparison Test is another possibility. Here we need a positive series that converges; we can use \sum\limits_{{n=1}}^{\infty }{{{{{\left( {\frac{1}{2}} \right)}}^{{n-1}}}}} again. We look at

\displaystyle \underset{{n\to \infty }}{\mathop{{\lim }}}\,\frac{{{{{\left( {1/3} \right)}}^{n}}}}{{{{{\left( {1/2} \right)}}^{n}}}}=\underset{{n\to \infty }}{\mathop{{\lim }}}\,{{\left( {\frac{2}{3}} \right)}^{n}}=0  and since the series in the denominator converges, our series converges absolutely.

So, for this example all the convergences that may be tested on the AP Calculus BC exam may be used with the single exception of the p-series Test which cannot be used with this series.

Teaching suggestions

  1. While the convergence of the series used here can be done all these ways, other series lend themselves to only one. Stress the form of the series that works with each test. For example, the Limit Comparison Test is most often used for rational expressions with the numerator of lower degree than the denominator and for expressions involving radicals of polynomials. The comparison is made with a p-series of whatever degree will make the numerator and denominator the same degree allowing the limit to be found.
  2. Most textbooks, after explaining each test and giving exercises on them, include a series of mixed exercises that require all the test covered up to that point. A good way to use this set is to assign students to state which test they would try first on each series. Discuss the opinions of the class and work any questions that students are unsure of or on which several ways are suggested.
  3. Give your students the series above, or a similar one, and have them prove its convergence using each of the convergence tests as was done above.
  4. Divide your class into groups and assign each group the series and one of the convergence tests. Ask them to use the test to prove convergence and then discuss the results as a group.

Of course, I didn’t really answer the question, did I? Check What Convergence Test Should I use Part 2

Updated February 23, 2013

Sequences and Series

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AP Type Question 10

Sequences and Series – for BC only

Convergence tests for series appear on both sections of the BC Calculus exam. In the multiple-choice section students may be asked to say if a sequence or series converges or which of several series converge.

The Ratio test is used most often to determine the radius of convergence and the other tests to determine the exact interval of convergence by checking the convergence at the end points. Click here for a Convergence test chart students should be familiar with; this list is also on the resource page.

On the free-response section there is usually one full question devoted to sequences and series. This question usually involves writing a Taylor or Maclaurin polynomial for a series.

Students should be familiar with and able to write several terms and the general term of a series. They may do this by finding the derivatives and constructing the coefficients from them, or they may produce the series by manipulating a known or given series. They may do this by substituting into a series, differentiating it or integrating it.

What Students Should be Able to Do 

  • Use the various convergence tests to determine if a series converges. The test to be used is rarely given so students need to know when to use each of the common tests. For a summary of the tests click: Convergence test chart.
  • Write the terms of a Taylor or Maclaurin series by calculating the derivatives and constructing the coefficients of each term.
  • Distinguish between the Taylor series for a function and the function. Do NOT say that the Taylor polynomial is equal to the function; say it is approximately equal.
  • Determine a specific coefficient without writing all the previous coefficients.
  • Write a series by substituting into a known series, by differentiating or integrating a known series or by some other algebraic manipulation of a series.
  • Know (from memory) the Maclaurin series for sin(x), cos(x), ex and \displaystyle \tfrac{1}{1-x} and be able to find other series by substituting into them.
  • Find the radius and interval of convergence. This is usually done by using the Ratio test and checking the endpoints.
  • Be familiar with geometric series, its radius of convergence, and be able to find the number to which it converges, \displaystyle {{S}_{\infty }}=\frac{{{a}_{1}}}{1-r}. Re-writing a rational expression as the sum of a geometric series and then writing the series has appeared on the exam.
  • Be familiar with the harmonic and alternating harmonic series.
  • Use a few terms of a series to approximate the value of the function at a point in the interval of convergence.
  • Determine the error bound for a convergent series (Alternating Series Error Bound and Lagrange error bound). See my post of  February 22, 2013.
  • Use the coefficients (the derivatives) to determine information about the function (e.g. extreme values).

This list is quite long, but only a few of these items can be asked in any given year. The series question on the exam is usually quite straightforward. As I have suggested before, look at and work as many past exam questions to get an idea of what is asked and the difficulty of the questions. Click on Power Series in the “Posts by Topic” list on the right side of the screen to see previous posts on Power Series.