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MATH 1321-004 Homework #1 due 2013-JAN-18, 11:35 a.m. 1 2 Guidelines Assignments For problems of determining convergence of series, a sim- All numbers below refer to the textbook: ple answer of convergence or divergence will not get any Calculus Concepts and Contexts, J. Stewart, 4th Ed. credit. One way to get full credit is to follow the tem• §8.2: 12, 16, 17, 20, 22, 30 plates illustrated as follows. P∞ ln n • §8.3: 2, 8, 10, 13, 20, 22, 29, 30, 31, 33, 41 Example 1.1 (Integral test). prove n=1 n diverges. Proof. ∀x ∈ [1, ∞), f (x) = ∃N = 3, s.t. ∀x ∈ [N, ∞), ln x x • §8.4: 6, 14, 21, 30, 34 satisfies f (x) ∈ (0, +∞). 1 − ln x f 0 (x) = < 0. x2 • Appendix D: 11, 23 (1) Boldfaced problems weigh 2 points each and all other problems 1 point each. A random subset of the assigned problems in §8.2, §8.3, and §8.4 with a total of 15 points Furthermore, will be graded. The two problems in Appendix D will t Z ∞ Z t ln x ln x (ln x)2 count as extra credit in addition to the 15 points. Addx = lim dx = lim = +∞. ditional extra credit of 10% will be given to you if you t→∞ t→∞ x x 2 1 1 1 (2) typeset your solutions in LATEX(ask me for the template P∞ ln n tex file if you plan to do this). n=1 n therefore diverges by the integral test. Caution: please make sure that your handwriting is Example 1.2 (Comparison test). determine whether recognizable, otherwise you only get partial credit for the P∞ 5 n=1 2n2 +4n+3 converges. recognizable part. 5 Solution. Let an = 2n2 +4n+3 , bn = 2n5 2 .P∀n ≥ 1, P an ≤ bn , it P is also clear thatP an > 0, bn > 0. P ∞ ∞ ∞ 5 1 b = , hence b converges by the 2 n n n=1 n=1 n n=1 2 P∞ convergence theorem on p-series. Therefore n=1 an converges by the comparison test. These examples are not intended for ‘copy-paste-andmodify’ usage; instead, try to bind your understanding to the math language while you write up the solution. You are expected to give similar solutions to at least one problem of the same type without referring to these templates. For other types of convergence tests, you need to 3 Tips The four steps of learning are: L1. understand the ideas, e.g. why the theorems hold, L2. memorize the packaging, i.e. the summary handout, L3. apply the math to solve problems, L4. integrate new learning into your knowledge system. Homework assignments will focus on theories while lab quizzes on real-world applications. Doing homework is in the early stage of L3, so it is often time-efficient to do step L2 before attempting the homework problems. However, it is also fine if you feel comfortable to try a few problems before step L2. Ask yourself two questions after finishing a problem: (a) identify the name of the test, (b) verify that all conditions (both the textbook version or my version are acceptable) hold, (c) give reasonable intermediate steps such as those shown in (2), • “What previous knowledge have I used?” (d) do (a) and (b) for each test if theorems of multiple tests are used. • “What is the connection between the previous knowledge to the new concepts and theorems?” 1