Why Convergence Tests?

Posted on February 6, 2024 by Lin McMullin

A large amount of time in Unit 10 is devoted to convergence tests. These tests tell you under what conditions a series will converge, when the infinite sum will approach a finite number.

The tests are really theorems. As with all theorems, you should learn and understand the hypotheses. This summery of the convergence tests lists the hypotheses of the tests that you are expected to know for the AP Calculus BC exam. The conclusion (at the top) is always that the series will converge or will not converge. You will likely spend a day or two on each test, learning how and when to use it. Use the summary to help you.

Some series have both addition and subtraction signs between the terms (often alternating). A series is said to be absolutely convergent or to converge absolutely if the series of absolute values of its terms converges. In effect, this means you may determine convergence by ignoring the minus signs. If a series converges absolutely, then it converges. This is an important way that many alternating series and series with some minus signs may be tested for convergence. If a series does not converge absolutely, it may still converge. In this case the series are said to be conditionally convergent.

Your goals is to learn which test to use and when to use it. The short answer is that you may use whichever test works. There is often more than one. Here are two blog posts discussing this. Read these after you’ve learned the convergence tests (but before your teacher’s test). The first post shows how different tests may be used on the same series. The second post gives hints on which test to try first. The key is the standard advice: Practice. Practice. Practice.

Course and Exam Description Unit 10, Sections 10.2 to 10.9. This is a BC only topic.

Why Power Series?

Posted on January 30, 2024 by Lin McMullin

The polynomial function $\displaystyle f\left( x \right)=x-\tfrac{1}{6}{{x}^{3}}+\tfrac{1}{{120}}{{x}^{5}}$ approximates the value of $\displaystyle \sin \left( {\tfrac{\pi }{6}} \right)$ correct to 5 decimal places:

$\displaystyle f\left( {\tfrac{\pi }{6}} \right)\approx 0.500002$

$\displaystyle \sin \left( {\tfrac{\pi }{6}} \right)=0.5$

This is not a fluke!

The graph of f(x) is in blue, the sin(x) in red. Note how close the two graphs are in the interval [-2, 2]

Now, approximating the value of a sine function is easier with a calculator. But sines are not the only functions in Math World.

In the Unit 10 you will learn how to write special polynomial functions, called Taylor and Maclaurin polynomials, to approximate any differentiable function you want to as many decimal places as you need. You already know a lot about polynomials. They are easy to understand, evaluate, and graph. The concept of using a polynomial to approximate much more complicated functions is very powerful.

You’ve already got a start on this! Recall that the local linear approximation of a function near x = a is $\displaystyle f\left( x \right)\approx f\left( a \right)+{f}'\left( a \right)\left( {x-a} \right)$ . This is a Taylor Polynomial. And it is the first two terms all the higher degree Taylor polynomial for f near x = a.

To fully understand these polynomials, there is a fair amount of preliminary stuff you need to understand. First you study sequences – functions whose domains are whole numbers. Next comes infinite series. A series is written by adding the terms of a sequence. (Sequences and series may have a finite or infinite number of terms. There is not much to say about finite series; infinite sequences and infinite series are where the action is.) oThe terms 0f some sequences and series are numbers. Other series have powers of an independent variable; these are called power series.

Some power series approximate (converge to) the related function everywhere (i. e. for all Real numbers). Others provide a good approximation only on an interval of finite length. The intervals where the approximation is good is called the interval of convergence. Convergence tests – theorems really – help you determine if a series converges. These in tern help you find the interval of convergence. More on this in my next post.

Depending on your textbook and your teacher, you may study these topics in this order: sequences, convergence test, series, Taylor and Maclaurin polynimials for approximations, and power series. Others may change the order. The path may be different, but the destination will be the same.

Course and Exam Description Unit 10, Sections 10.1, 10.2, 10.11, 10.13, 10.14, 10.15. This is a BC only topic.

Unit 10 – Infinite Sequences and Series

Posted on January 25, 2022 by Lin McMullin

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 n^th 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 e^x.

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

Adapting 2021 BC 6

Posted on August 31, 2021 by Lin McMullin

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

Posted on January 19, 2021 by Lin McMullin

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.