Last updated Sat Apr 1 00:43 EDT 2017
Due-date: Friday 4/7/17
You are required to do all of the problems below. You will not be required to hand them all in. I have indicated below which ones you do have to hand in on the due-date. Don't make the mistake of thinking that I'm collecting only the problems I think are important.
The "due date" above is the date that your written-up problems should be handed in, but don't wait to get started on the assignment. You should always get started on problems as soon as we cover the relevant material in class.
- A. Rosenlicht Chap. VII (pp. 160-167)/ 3–5, 9, 10. Of these, hand in only 3 and 9. In these exercises, remember that for a sequence of functions, Rosenlicht's "converges" is my "converges pointwise" (and similarly for "convergent").
- B.
- In the handout on improper integrals, do exercises 1–8, 10, 12, 16. (I have reduced the original "1–16" assignment. However, all of the problems 1–16 are still fair game as source-material for exam problems.) Note that exercise 1 in this handout is essentially the formerly-optional Exercise 6.11 on p. 34 of the Notes on Integration (Mar. 22 draft). Of these, hand in only 1--4, 6, 10. Within the Improper Integrals handout, when doing any problem you may assume the results of all earlier problems in the handout, whether or not assigned. In particular, in #6 you may use the relevant analog referred to in #5 (which previously had a typo that's now been corrected). For #15 (which is no longer part of this assignment, but I'm keeping the following information here anyway), use the following definition: $${\lim\,\inf}_{x\to\infty} f(x) := \lim_{x\to\infty}(\inf\{f(y): y\geq x\}).$$ (For any function \(f:[a,\infty)\to {\bf R}\) this limit exists in the extended reals, \({\bf R}\cup\{\pm\infty\}\), since the function \(x\mapsto \inf\{f(y): y\geq x\}\) is monotone increasing.) For real-valued sequences \((a_n)\), we define \(\lim\inf_{n\to\infty} a_n\) similarly, replacing the domain-variable \(x\) by the natural number \(n\). We also define \(\lim\sup_{x\to\infty} f(x)\) and \(\lim\sup_{n\to\infty} a_n\) analogously.
- In Section 6.9 of the Notes on Integration, do Exercises 6.12–6.15. Of these, hand in only 6.12.
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