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.