Last updated Thu Oct 28 21:32 EDT 2021
Due-date: Wed., 11/3/21
You are required to do all of the problems and reading below (except for anything explicitly labeled "optional"). 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 are to 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: Abbott exercises. See notes and additional instructions below this list before starting these problems.
Of the Abbott problems, hand in only these:
- 2.2/ 2, 4ab
- 2.3/ 1, 2, 4, 5, 7, 8
- 2.4/ 1–3, 7.
- 2.3/ 1, 5, 8
- 2.4/ 1, 3a, 7d. In your write-up for 7d, you may assume the results of 7abc.
Notes and additional instructions on some of these problems:
- 2.2.4(a) and (b): The words "ones" and "one" should be replaced by "1's" and "1". Abbott's use of the plural "ones" in (a) and (b) is not terrible, but using the word "one" at the end of the sentence (in both (a) and (b)) is poor, because "one" is often used as a pronoun. The reader could reasonably ask, "One what?" and would need to re-read the sentence to see that there's no noun to which this last "one" could refer. And if the reader, new to the subject, didn't know that a sequence can't converge to another sequence, he or she might think that this last "one" could refer to an earlier noun, namely "sequence".
Abbott appears to be mis-applying a rule you may have learned in high school, "Spell out numbers [non-negative integers] that are at most twenty." That rule is intended for numbers that refer to some count, not for real numbers that happen to be integers. Nobody in his right mind would write the interval \( \{x\in {\bf R}: 1\leq x\leq 2\}\) as "[one,two]."
An example in which such usage of "one" would lead to serious ambiguity: "There are infinitely many numbers in the interval [0,2], and our sequence \( (a_n)_{n=1}^\infty\) converges to one."
Also, it would be better to say "infinitely many 1's" rather than "an infinite number of 1's." The latter is baby-talk aimed at students below the level of this course, and uses the word "number" in a questionable way that can easily be avoided (and therefore should be avoided).
- 2.3.1(b) (i) Change the wording to: "If \( (x_n)\to x\ \), show that \(x\geq 0\) and that \( \sqrt{x_n}\to \sqrt{x}\)."
(ii) A useful identity for this problem is \(\sqrt{a}-\sqrt{b}= \frac{a-b}{\sqrt{a}+\sqrt{b}}\) , valid whenever \(a,b \geq 0\) and at least one of \(a,b\) is strictly positive.- 2.3.7. For part(s) in which "such a request is impossible", you may need to use "the proper theorem(s)", not merely reference them.
- 2.3.8
- The statement of this exercise violates two rules of mathematical writing: (i) the same letter \(x\) is used both as a fixed number (the limit of the sequence \( (x_n)_{n=1}^\infty \) ) and as a variable (in "\(p(x)\)"), and (ii) the name of a function (in this case \(p\) and \(f\) ) does does not include notation for the domain-variable (the functions in this problem are \(p\) and \(f\); even were the letter \(x\) not already in use as the name of a specific real number, these functions should not be denoted \(p(x)\) and \(f(x)\) and more than they should be denoted \(p(y)\) and \(f(y)\), or \(p({\rm giraffe})\) and \(f({\rm hippo})\).
To fix these problems:
- In the first sentence of the exercise, replace "let \(p(x)\) be a polynomial" with "let \(p:{\bf R}\to {\bf R}\) be a polynomial function" (but do not change anything in part (a)).
- In part (b), replace the first "\(f(x)\)" with " \(f:{\bf R}\to {\bf R}\)" (but do not change the second "\(f(x)\)").
- In part (a), keep in mind that we have not defined "continuous function", or limits of functions of a real variables, yet. Do this exercise by repeated applications of our "Limits of sequences `behave well' with respect to arithmetic" result.
- 2.4/1–3.
- Do non-book problem B4 first.
- It is no accident that these exercises were placed in this section of the book. Write out the first few terms of each sequence (this is already done for you in 2.4.3). With the the exception of the sequence in 2.4.2(a), you should notice that based on the first few terms, it appears that each of these sequences is monotone and bounded. If you can prove that a given sequence is, indeed, monotone and bounded (which can't be done just by examining a finite number of terms and writing "..." !), then you can apply a result from class to show that that sequence converges. Note that if \( (a_n)_{n=1}^\infty\) converges, then \( \lim_{n\to\infty} a_{n+1}=\lim_{n\to\infty} a_n\). (This is a special case of non-book problem B4.)
- In each of these problems, if the given sequence is \( Z:=(z_n)_{n=1}^\infty \), then after choosing a value for \( z_1 \), the other terms are defined recursively by an equation of the form \( z_{n+1} = f(z_n) \), where \(f\) is either a rational function or the square root of a rational function. (Recall that a a rational function is polynomial function divided by another polynomial function. More precisely, a [real-valued] rational function of a real variable is function \(g\) for which (i) there is a polynomial function \(q:{\bf R}\to {\bf R}\) such that the domain of \(g\) is the set \(D=\{x\in{\bf R}: q(x)\neq 0\}\), and (ii) there is another polynomial function \(p:{\bf R}\to {\bf R}\) such that \(g(x)=\frac{p(x)}{q(x)}\) for all \(x\in D\). Polynomial functions are, themselves, special cases of rational functions; we simply take the denominator of the rational function to be the constant function 1 [a polynomial function of degree zero]). Note that we have not yet defined "continuous function", or limits of functions of a real variables. However, with repeated applications of "sequences in \({\bf R}\) behave well with respect to arithmetic"—several of which can be avoided in this problem if you've already done problem 2.3.8(a)— and (where relevant) exercise 2.3.1(b), you should be able to show that for any function \(f\) as above (or at least for the \(f\)'s in these exercises), if \(Z\) converges then \(\lim_{n\to\infty} f(z_n) = f(\lim_{n\to\infty} z_n). \) But note that, in the case of a rational function with nonconstant denominator, you will need to show that limit of the denominator is nonzero in order for \( f(\lim_{n\to\infty} z_n) \) even to be defined.
- 2.4.1(c) Carefully justify all your steps.
- 2.4.3:
- Prove your answers.
- In each part of this problem, you have to start by carefully defining the sequence—say \( (x_n)_{n=1}^\infty\)—recursively. I.e. you need to find a function \(f\) so that the terms beyond the first are given by the equation \(x_{n+1}=f(x_n)\). Everything you prove will be based on this relation, not on the visual appearance of the terms on the page. For example, it may appear "obvious" to you that the sequence is monotone increasing, but once you figure out \(f\) you'll realize that the monotonicity isn't as obvious as you thought.
- 2.4.7(b):
- The "reasonable definition" is, of course, obtained by replacing "sup" with "inf" wherever "sup" appears earlier in the exercise. I would have used the word "analogous" rather than "reasonable".
- As you might guess from the terminology "limit superior", the abbreviation \(\liminf\) stands for "limit inferior", not "limit infimum".
- 2.4.7(d): Reword second sentence as: "In this case, show that all three limits have the same value."
- B: Click for non-book problems. Of these, hand in only B3, B8.
- C: In Abbott, read Example 2.4.5, Theorem 2.4.6 (including the proof), and Corollary 2.4.7.
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