MATH 3100: Sequences and series

Spring 2013

MWF 1:25-2:15 PM, Boyd Graduate Studies Building, Room 303
Instructor: Paul Pollack

Current assignments/other course material

• Required: Assignment #11, due Friday, April 26 at the start of class
  Extra problem: Assume $f$ and $g$ are real-valued functions which have $n$ derivatives at $0$, and that all of their first $n$ derivatives are continuous at $0$. In this exercise, we walk through the proof of the rule $P_n^{fg} = [P_n^{f} P_n^{g}]_n$.
  1. Suppose that $P$ is a polynomial which agrees with $f$ to order $n$ at $0$, and that $Q$ is a polynomial which agrees with $g$ to order $n$ at $0$. Show that the polynomial $PQ$ agrees with $fg$ to order $n$ at $0$. Hint: Rewrite the difference $PQ-fg$ as $g(P-f)+f(Q-g)+(P-f)(Q-g)$.
  2. Deduce that $P_n^{f} P_n^{g}$ agrees with $fg$ to order $n$ at $0$.
  3. Deduce that $[P_n^{f} P_n^{g}]_n$ also agrees with $fg$ to order $n$.
  4. Use the uniqueness part of Theorem 3.2.5 to conclude that $[P_n^f P_n^g]_n = P_n^{fg}$.
  §3.2: Exercises 3, 4, 5, 8 [interpret `within 5 decimal places' to mean within $10^{-5}$], 10, 11
  

  Recommended exercises (not to turn in):
  §3.2: Exercises 1, 2, 7, 14, 15
  

• Assignment #10, not to be collected
  Extra problem: Let $R$ be the radius of convergence of the power series $a_0 + a_1 x + a_2 x^2 + \dots$, and let $R'$ be the radius of convergence of $a_1 x + 2a_2 x^2 + 3a_3 x^3 + \dots$. In this exercise, we outline a proof that $R=R'$. It suffices to show that the series $\sum_{n=0}^{\infty} n a_{n} x^{n}$ converges for $|x| < R$ and diverges for $|x| > R$. Prove this as follows:
  1. Show that if the series $\sum_{n=0}^{\infty} n a_n x^{n}$ converges absolutely for a particular $x \in \mathbb{R}$, then $\sum_{n=0}^{\infty} a_n x^{n}$ converges absolutely for the same value of $x$.
  2. Suppose for the sake of contradiction that $\sum_{n=0}^{\infty} n a_n x^{n}$ converges when $x=c$ for some real number having $|c| > R$. Derive a contradiction from part (a) and Proposition 2.4.3.
  3. Now take a real number $x$ with $|x| < R$. Show that $\sum_{n=0}^{\infty} n a_n x^{n}$ converges absolutely. Hint: Let $c$ be a real number between $|x|$ and $R$. Compare the series $\sum_{n=0}^{\infty} n |a_n| |x|^{n}$ to the series $\sum_{n=0}^{\infty} |a_n| c^n$.
  Strongly recommended exercises: §3.1: 4(a,c,d,e,f), 5, 6(a--d), 7, 10, 11
  Less strongly recommended exercises:: §3.1: 1, 2, 9, 12
• Required: Assignment #9, due Friday, April 5 at the start of class
  §2.4: Exercises 1(b,e,f,h,k,l), 2, 3, 5, 6
  

  Recommended exercises (not to turn in):
  §2.4: Exercises 1(a,c,d,g,i,j), 4
  

• Required: Assignment #8, due Friday, Mar. 29 at the start of class
  Do the following extra problem, which completes the proof of the ratio test: Suppose that $\sum_{n=1}^{\infty} a_n$ is an infinite series with each term $a_n \neq 0$. Suppose that $ \left|\frac{a_{n+1}}{a_n}\right|$ tends to either a finite real number $L > 1$, or diverges to $\infty$.
  1. Let $b_n = 1/a_n$. Show that $\lim_{n\to\infty} \left|\frac{b_{n+1}}{b_n}\right|$ exists and is $< 1$.
  2. Deduce that $\sum_{n=1}^{\infty} b_n$ converges absolutely.
  3. Explain why the result of (b) implies that $\lim_{n\to\infty} \frac{1}{|a_n|} =0$.
  4. Deduce that $\sum_{n=1}^{\infty}a_n$ does not converge.
  (Exercise 6 in §1.5 might be helpful for some of this.)
  Then do: §2.3: Exercises 2, 3(b,c,f,g), 6, 9, 11, 12
  

  Recommended exercises (not to turn in):
  §2.3: Exercises 1, 3(a,d,e), 7, 8, 10
  

• Strongly recommended: Assignment #7 (not to be collected)
  Do both of the following:
  1. Suppose that $\sum_{n=1}^{\infty}a_n$ and $\sum_{n=1}^{\infty}b_n$ are series with nonnegative terms, and that $\{a_n/b_n\}$ diverges to infinity. Show that if $\sum_{n=1}^{\infty}b_n$ diverges, then $\sum_{n=1}^{\infty} a_n$ also diverges. [This is the part of the limit comparison that we didn't prove in class.]
  2. First, show that for every natural number $j$, we have $\frac{1}{j+1} \leq \int_{j}^{j+1} \frac{dx}{x} \leq \frac{1}{j}$. Deduce that \[ 0 \leq \frac{1}{j} - \int_{j}^{j+1}\frac{dx}{x} \leq \frac{1}{j(j+1)}. \] Use this and the comparison test to show that \[ \sum_{j=1}^{\infty} \left(\frac{1}{j} - \int_{j}^{j+1}\frac{dx}{x}\right) \] converges. [Recall that the sum of this series is what we called Euler's constant $\gamma$.]

  Then do:
  §2.2: Exercises 1, 2, 3, 4, 6
  

  Less strongly recommended exercises:
  §2.2: Exercises 5, 7
  

• Required: Assignment #6, due Friday, Mar. 1 at the start of class
  §2.1: Exercises 2, 4, 9, 12, 13, 14, 15
  

  Recommended exercises (not to turn in):
  Read the lyrics and/or listen to the MP3 of Tom Lehrer's song "There's a delta for every epsilon". Then do:
  §2.1: Exercises 1, 3, 5, 6, 8, 10
  

• Required: Assignment #5, due Friday, Feb. 22 at the start of class
  §1.6: Exercise 2 (note the hint at the top of the p. 70), 5, 8, 11
  §1.7: Exercise 1, 4, 7

  Recommended exercises (not to turn in):
  §1.6: Exercises 6, 7, 9, 10, 12
  §1.7: Exercises 2, 3, 5, 6

• Strongly recommended (but not to turn in): Assignment #4, solutions to be posted by me by Feb. 6

  §1.5: Exercises 1, 3, 4, 5, 6, 7(a) [we proved the squeeze theorem slightly differently in class], 9. You might prefer to wait until after Monday's lecture to attack problems 1 and 6.

  Less strongly recommended exercises (not to turn in): 10, 12

• Required: Assignment #3, due Friday, Jan. 31 at the start of class

  On Monday, 1/28, we will prove Proposition 1.4.15 about geometric sequences in the special case when the common ratio $r > 0$. The following exercise asks you to take over and do the cases when $r < 0$. (The approach in the book is a little different, so in your solution you shouldn't just copy the book's proof of Proposition 1.4.15.)

  Exercise: Show that if $-1 < r < 0$, then $\{r^n\}$ converges to $0$, while if $r \leq -1$, the sequence $\{r^n\}$ diverges.

  Then do: §1.4: Exercises 2 [if you like, you can replace the condition $n > N$ with $n \geq N$ as in class; just be clear which version you are doing!], 8, 10, 15, 17, 23

  Recommended exercises (not to turn in): 1, 3, 4, 5, 6, 7, 11, 16, 20, 21, 22

• Required: Assignment #2, due Friday, Jan. 25 at the start of class

  First, do the following exercise, which is a version of §1.3, Exercise 12:

  Exercise: Show that the sequence $\{b_n\}$ with $b_n = 2^n$ is a subsequence of the sequence $\{a_n\}$ defined by $a_n=2n$ by exhibiting a strictly increasing function $g\colon \mathbf{N}\to \mathbf{N}$ with $b_n = a_{g(n)}$ for all $n \in \mathbf{N}$.

  Then do §1.3: Exercises 8, 9 (no proofs necessary), 13, 15, 21.

  Recommended exercises (not to turn in): 10, 14, 17, 18, 19, 20, 25

• Assignment #1, due Friday, Jan. 18 at the start of class
  All problem references are to the latest version of the notes, available at <URL: http://www.math.uga.edu/~adams/SANDS13.pdf>
  §1.1: 4, 5, 8
  §1.2: 2, 11, 14(b), 18, 20

  The argument in problem 20 is perhaps not stated in the most convincing form. Whatever refutation you propose should apply to the following (somewhat clearer) version of the argument: Let $P(n)$ be the statement that any group of n cows have the same color. Clearly $P(1)$ is true. Now suppose $P(n)$ is true for some natural number $n$, and let's prove $P(n+1)$. To prove $P(n+1)$, we have to show that any group of $n+1$ cows has the same color. Take any group of $n+1$ cows, say $C_1, ..., C_{n+1}$. Since $P(n)$ holds, the first $n$ cows $C_1, \dots, C_n$ have the same color, and similarly for the last $n$ cows $C_2, \dots, C_{n+1}$. But these two group of cows overlap; for instance, the second cow $C_2$ is on both lists. So all of our $n+1$ cows have the same color (namely, the color of the cow $C_2$). Since this argument holds for any collection of $n+1$ cows, the statement $P(n+1)$ holds.

Other course materials:

• A proof that a certain sequence is strictly increasing [PDF]; this proof may serve as a model for your own HW write-ups

The UGA bulletin describes the content of this course as follows:

Precise definitions of limit and convergence concepts; practical tests for convergence of infinite series; power series representations and numerical error estimates; applications to calculus and explicit summation formulae; trigonometric series.

This isn't wrong (although we might not cover all of these topics), but it fails to capture the spirit of the course. You should view this class as a re-introduction to sequences and series (and to the real numbers). You have surely met these topics in calculus, but the rushed pace of a calculus course makes it doubtful that your introduction got off on the right foot. We will move at a more leisurely pace, with an emphasis on developing a deep understanding of what's `really going on'. It is hoped that by the end of the course, any residual SSAD (Sequences and Series Anxiety Disorder) will have vanished, allowing you to look ε's, δ's , and N₀s straight in the eye.

A second raison d'être of this course is that you continue the mathematical apprenticeship which you begun in MATH 3200. This includes the construction of carefully reasoned mathematical proofs. You should not expect this to be easy; indeed, as you mature as a mathematician, you will find yourself confused a good deal of the time! (By the time you get to be a professional mathematician, you are constantly confused.) I will do my best to guide you through the difficulties and to help you come out the other side. Of course, this depends a great deal on your own engagement with the material -- both in class and in office hours.

We will use the course notes by UGA's own Professor Malcolm Adams, which are freely available online at <URL: http://www.math.uga.edu/%7Eadams/SANDS13.pdf>. These notes were written with this course in mind, and so we will aim to make it all the way through.

Instructor: Paul Pollack E-mail: pollack at math dot uga dot edu

Office: 318 Boyd Graduate Studies Building

Office hours: Monday, 2:30--3:25 PM Wednesday, 2:30--3:30 PM Thursday, 2--3:00 PM

Exam dates

There are three in-class midterm exams as well as a final exam.

• Midterm #1: Friday, Feb. 8
• Midterm #2: Friday, March. 8
• Midterm #3: Friday, April 12
• Final exam: Friday, May 3, 12:00 -- 3:00 pm (usual classroom)

Your grade is made up of the following weighted components:

• Each midterm: 16% (total of 48%)
• Homework: 22%
• Final exam: 30%

You are expected to participate in class. In particular, attendance in this course is required. More than four unexcused absences may result in you receiving a WF. Keep me posted whenever you have a conflict that requires you to miss class and this should not be an issue.

All exams are in-class, closed book and closed notes.

Homework will be collected in class, about once each week. Late homework will not be accepted. (If you have a need to turn in HW early, that can be arranged.) Your lowest HW score will be dropped at the end of the term.

On homework, collaboration is allowed and in fact is very much encouraged. Mathematics wouldn't be nearly as much fun if we couldn't talk about it with other people! However, However, copying (from a textbook or another student) and web searches are not allowed, and you must write your own final solutions independently. Keep in mind that by entering UGA, you have already agreed to abide by the UGA Honor code described in detail at <URL: http://honesty.uga.edu/ahpd/culture_honesty.htm>.

In practice, what this means that you may discuss homework problems and their solutions with your classmates, but you may not turn in a solution unless you understand it yourself. A reasonable rule of thumb is that you should be able to explain your solutions verbally to me (in all their gory detail) if requested to do so.

The withdrawal policy for this course is that if you withdraw within a week of the date the first midterm is handed back, you automatically qualify for a WP (assuming you are eligible by UGA's standards). After that point, it is at the discretion of the instructor (i.e., me) who will take your performance in the class to-date into account.

Students with disabilities who may require special accommodations should talk to me as soon as possible. Appropriate documentation concerning disabilities may be required. For further information, please visit the Disabilities Resource Center page at <URL: http://drc.uga.edu/>.