Last updated Wed Dec 9 20:38 EST 2020
Due-date: day of final exam (Wed. Dec. 16), but there is nothing to hand in.You should do all of the problems below. You will not hand in anything. The purpose of the new exercises on this assignment is to give you practice with material covered since Assignment 5. (Part B also includes some older work exercises that you may have neglected to do.)
- A: Bartle & Sherbert exercises:
- Section 3.7 (yes, 3.7)/ 3b, 5, 7, 10, 11, 12
- Section 9.1/ 7, 9
- Section 9.2/ 1cd, 2abd
- Section 9.4/ 1ab, 2
Notes on some of these problems:
- Note on 3.7/ 10: Obviously, you should first read Example 3.7.6(f) (p. 98, the alternating harmonic series).
- Note on 3.7/ 11: Consider the book's problem to be just part (a). For part (b), remove the hypothesis that \(a_n>0.\)
- Note on 9.4/1ab: Interpret the instruction to "discuss" the convergence and uniform convergence of the indicated series to mean this: (i) Find the "domain of convergence", i.e. the set of all \(x\in{\bf R}\) for which the series converges. (ii) Determine whether the convergence is uniform on the domain of convergence. (iii) If the convergence is not uniform on the whole domain of convergence, find all closed intervals on which the convergence is uniform. (Remember that there are unbounded closed intervals as well as bounded closed intervals.)
- Note on 9.4/2: Insert the words "Prove that" at the beginning of the problem.
- B: For anything in the lecture notes that was labeled an exercise, or HW, or was referred to with words like "Left to student": do the indicated exercise/HW or fill in the omitted details, if you haven't already done so. If you've been studying properly all along, then the only new work should be for recent lectures, but you should look back to see if there's anything you forgot to do in older lecture notes.
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