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Name: Group: MATH 104 SAIL, Fall 2018 Convergence Tests

Snapshot

• Major Concept: There are many tests that can help determine convergence or of a or . It’s important to know what conditions each test requires and when each test is conclusive or inconclusive. • Before You Begin: Briefly review all the tests from sections 10.2–10.6. • Standards for Practice and Evaluation: Like the integration techniques, you’ll need to develop your own intuition about when and how to apply each test (i.e., it’s unlikely that you’ll be told which test to use). Devote particular energy to deciding which tests are the most efficient (easiest) in particular contexts and when tests would be inconclusive.

Warm-up: Direct Application

Worksheet Objective

Practice direct application of each of the tests we have learned. Here and on exams, verify that the given test actually applies unless you are explicitly instructed to ignore such issues.

Remember

• Remember that the most important part of doing worksheets is having finished worksheets. • As always, use pencil or scratch paper first when working these questions so that you can save your future self from revisiting the gory details of ideas that might not have worked. • You might also want to also leave warnings about the intuition and impulses that didn’t work to save your future self from duplicating ideas that might not have worked! • Remember for the questions below that it is unlikely that an exam question would tell you specifically which test to use.

Remember Understand Apply Analyze Evaluate Create

Use the n-th Term Divergence Test to show that Use the Test to determine convergence or diver- ∞ ∞ X 2n X diverges. gence of the series ne−n. 2n + 1 n=1 n=1

November 6–8, 2018 Worksheet 15–1 Name: Group: MATH 104 SAIL, Fall 2018

Remember Understand Apply Analyze Evaluate Create

Use the Comparison Test and compare the series Use the and compare the series ∞ ∞ X 1 X 2k to a p-series to determine convergence or di- to a to determine conver- n5 + 1 3k+1 − k n=3 k=0 vergence. gence or divergence.

∞ X (−5)n Use the Test to show the series Use the to show that the series n di- ∞ 3 2 3 + n X (−1)n +3n +5 n=1 converges. verges. n5 n=1

∞ X 2 Use the to show that the series (−1)ne−n Use the Test to determine whether the ∞ n=3 X k + 1 converges. series (−1)k converges or diverges. k k=3

November 6–8, 2018 Worksheet 15–2 Name: Group: MATH 104 SAIL, Fall 2018

Choosing Tests and Comparisons

Remember Understand Apply Analyze Evaluate Create

Here and on the next page, use the tests from the previous page to determine whether or not the series converges. State which test(s) you’ve used, how you used it, and its conclusion. Questions will be split among the groups; your group will receive instructions about which ones it should work first.

∞ ∞ 2 ∞ 2 X 1 X (−1)n +2n X (−1)n +2n n2 + 1 n2 + 1 n2 − 1 n=1 n=1 n=2

∞ ∞ ∞ X ln n X (ln n)3 X (ln n)2 n3 n n2 n=1 n=1 n=1

∞ ∞ ∞ X 1 X 1 X n 2n − 1 2n − n 2n − n n=1 n=1 n=1

November 6–8, 2018 Worksheet 15–3 Name: Group: MATH 104 SAIL, Fall 2018

Remember Understand Apply Analyze Evaluate Create

∞ ∞ ∞ X (−1)n X (−1)n ln n X (−1)n ln n (ln n)2n 2n 2n − (1.5)n n=2 n=1 n=1

∞ ∞ ∞ X 2n X nn X 2n √ n! n! n=1 n=1 n=1 n!

∞ ∞ ∞ X x n X x n X (−1)n (assume |x| < 1) (assume |x| > 1) n2 n2 n2 n=1 n=1 n=1

November 6–8, 2018 Worksheet 15–4 Name: Group: MATH 104 SAIL, Fall 2018 Convergence Tests III: Advanced Applications

Worksheet Objective

On this worksheet, you will practice some particularly challenging applications of the convergence tests we have studied. Note that a number of these questions are taken directly from old final exams.

Remember Understand Apply Analyze Evaluate Create Determine whether the series below are convergent or divergent. For each series, create a companion series which is similar to the given one but has the opposite behavior (divergent instead of convergent, etc.).

∞ 2 ∞ πn ∞ X 6n X cos X 1 1 2 sin n! n + 1 k k n=1 n=0 k=1

∞ 5 −j ∞  3 2 2n ∞ −n4 X j cos j + e sin j X n + n + 2 X  1  1 + j 7 − 1 3n3 − n2 + 1 n3 j=3 n=1 n=1

∞ ∞ n n ∞ X arctan n X 2 − 5 X 1 e ` n 3n + 4n n=1 n=0 `=1

November 6–8, 2018 Worksheet 15–5 Name: Group: MATH 104 SAIL, Fall 2018

Remember Understand Apply Analyze Evaluate Create

P∞ P∞ an Suppose that n=0 an converges to 2. Is the series n=1 e convergent or divergent?

Remember Understand Apply Analyze Evaluate Create

It is a strange but true fact that ∞ X 1 π2 = . n2 6 n=1 How many terms of the series would you need to add to check the accuracy of this equality up to ten decimal places?

It is also true that ∞ X (−1)n π2 = − . n2 12 n=1 How many terms of the series would you need to add to check the accuracy of this equality up to ten decimal places?

Review and Summary

As you continue your practice using convergence tests, focus on these elements:

• Look for problems which require you to make the choice of tests—this is most likely what you will be asked to do on exams. • Be particularly aware of the hypotheses and conditions that are required for a test to apply. A favorite exam question is to give an alternating series which does not meet the criteria necessary for the . • Also be particularly aware of the cases when a test is inconclusive. In such cases you’ll need to use a different test. Know which tests are similar enough (e.g., ratio and root) that it’s not worth trying both when one is inconclusive. • Finally, practice using orders of growth methods to evaluate limits quickly.

For next time, begin reading about in Section 10.7.

November 6–8, 2018 Worksheet 15–6