414/514 The theorems of Riemann and Sierpiński on rearrangement of series

November 16, 2014

I.

Perhaps the first significant observation in the theory of infinite series is that there are convergent series whose terms can be rearranged to form a new series that converges to a different value.

A well known example is provided by the alternating harmonic series,

\displaystyle 1-\frac12 +\frac13-\frac14+\frac15-\frac16+\frac17-\dots

and its rearrangement

\displaystyle 1-\frac12-\frac14+\frac13-\frac16-\frac18+\frac15-\dots

According to

Henry Parker Manning. Irrational numbers and their representation by sequences and series. John Wiley & Sons, 1906,

Laurent evaluated the latter by inserting parentheses (see pages 97, 98):

\displaystyle \left(1-\frac12\right)-\frac14+\left(\frac13-\frac16\right)-\frac18+\left(\frac15-\frac1{10}\right)-\dots \displaystyle=\frac12\left(1-\frac12+\frac13-\frac14+\dots\right)

A similar argument is possible with the rearrangement

\displaystyle 1+\frac13-\frac12+\frac15+\frac17-\frac14+\dots,

which can be rewritten as

\displaystyle 1+0+\frac13-\frac12+\frac15+0+\frac17+\dots \displaystyle =\left(1-\frac12+\frac13-\frac14+\frac15-\frac16+\frac17-\dots\right) \displaystyle +\left(0+\frac12+0-\frac14+0+\frac16+0-\dots\right) \displaystyle =\frac32\left(1-\frac12+\frac13-\frac14+\dots\right).

The first person to realize that rearranging the terms of a series may change its sum was Dirichlet in 1827, while working on the convergence of Fourier series. (The date is mentioned by Riemann in his Habilitationsschrift, see also page 94 of Ivor Grattan-Guinness. The Development of the Foundations of Mathematical Analysis from Euler to Riemann. MIT, 1970.)

Ten years later, he published

G. Lejeune Dirichlet. Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1837, 45-81,

where he shows that this behavior is exclusive of conditionally convergent series:

Theorem (Dirichlet). If a series converges absolutely, all its rearrangements converge to the same value.

Proof. Let u_0,u_1,\dots be the original sequence and u_{\pi(0)},u_{\pi(1)},\dots a rearrangement. Denote by U_0,U_1,\dots and V_0,V_1,\dots their partial sums, respectively. Fix \epsilon>0. We have that for any n, if m is large enough, then for all i\le n there is some j\le m with \pi(j)=i. Also, there is a k such that for all j\le m there is a i\le k with \pi(j)=i, so

|U_m-V_m|\le\sum_{i=n+1}^m|u_i|+\sum_{i=n+1}^k|u_j|.

Choosing n large enough, and using that \sum_i|u_i| converges,  we can ensure that the two displayed series add up to less than \epsilon. This gives the result. \Box

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314 – Foundations of Analysis – Syllabus

January 20, 2014

Math 314: Foundations of Analysis.

Andrés E. Caicedo.
Contact Information: See here.
Time: TTh 12:00 – 1:15 pm.
Place: Mathematics Building, Room 139.
Office Hours: Th, 1:30 – 3:00 pm, or by appointment. (If you need an appointment, email me a few times/dates that may work for you, and I’ll get back to you).

Textbook: Stephen Abbott. Understanding Analysis. Springer-Verlag, Undergraduate Texts in Mathematics, 2001; 257 pp. ISBN-10: 0387950605. ISBN-13: 978-0387950600.

Here is the publisher’s page. Additional information is available from the author’s page. Review (MR1807438 (2001m:26001)) by Robert Gardner Bartle at MathSciNet. Review by Jeffrey Nunemacher at the American Mathematical Monthly, Vol. 118, No. 2 (February 2011), pp. 186-189.

I will mention additional references, and provide handouts of additional material, as needed.

Contents: The department’s course description reads:

The real number system, completeness and compactness, sequences, continuity, foundations of the calculus.

I strongly suggest you read the material ahead of our meetings, and work on it frequently. You may find some of the topics challenging. If so, here is some excellent advice by Faulkner (from an interview at The Paris Review):

FaulknerPersonally, I find the topics we will study beautiful, and I hope you enjoy learning it as much as I did.

Please bookmark this post. I update it frequently with detailed week-to-week descriptions.

Detailed day to day description and homework assignments. All problems are from Abbott’s book unless otherwise explicitly specified:

  • January 21 – 30. Chapter 1. The real numbers. Irrationality. Completeness. Countable and uncountable sets.
  • January 21. Functions. Mathematical induction and the well-ordering principle.
  • January 23. Sets, logic, quantifiers. Completeness.
  • January 28. Completeness. Countable and uncountable sets. I recommend you read Errol Morris‘s essay on Hypassus of Metapontum, the apparent discoverer of the irrationality of \sqrt2.
  • January 30. Comparing infinities. Counting the rationals. I recommend the following two papers on this topic: 1 and 2. Office hours this week will be on Friday, 11:45-1:15.

Homework set 1 (Due February 4). Exercises 1.2.1, 1.2.2, 1.2.7, 1.2.8, 1.2.10; 1.3.2 – 1.3.9; 1.4.2 – 1.4.7, 1.4.11 – 1.4.13; 1.5.3, 1.5.4, 1.5.9. See below for the required format.

Solution to 1.2.1.

  • February 4 – 20. Chapter 2. Sequences and series. Limits. Cauchy sequences. Infinite series. Riemann‘s rearrangement theorem.
  • February 4. Rearrangements of infinite series, limits of sequences. Homework 1 is due today.
  • February 6. Limit theorems.
  • February 11. Limit theorems continued. Infinite series.
  • February 13. Monotone convergence. The Bolzano–Weierstrass theorem.
  • February 18. The Bolzano-Weierstrass theorem continued. Absolute and conditional convergence. Cauchy sequences.
  • February 20. Riemann’s rearrangement’s theorem, and extensions (see here and here). The interesting paper by Marion Scheepers mentioned on the second of those links can be found here.
  • Additional topics: Products of series. Double series.

Homework set 2 (Due February 25). Exercises 2.2.1, 2.2.2, 2.2.5, 2.2.7, 2.3.2, 2.3.3, 2.3.6, 2.3.7, 2.3.9, 2.3.11, 2.4.2, 2.4.4, 2.4.5, 2.5.3, 2.5.4, 2.6.1, 2.6.3, 2.6.5, 2.7.1, 2.7.4, 2.7.6, 2.7.9, 2.7.11. See below for the required format.

  • February 25 – March 6. Chapter 3. Basic topological notions: Open sets. Closed, compact, and perfect sets. The Cantor set. Connectedness. The Baire category theorem.
  • February 25. The Cantor set. Open and closed sets.
  • February 27. Open and closed sets, continued. Extra credit problem: Find a set of reals such that we can obtain 14 different sets by applying to it (any combination of) the operations of complementation and closure. Kuratowski showed that 14 is the largest number that can be obtained that way, you are welcome to also try to show that. (See here.)
  • March 4. Open covers, compact sets. Perfect sets. Connectedness.
  • March 6. The Baire category theorem.
  • Additional topics: The study of closed sets of reals naturally leads to the Cantor-Bendixson derivative, and the Cantor-Baire stationary principle (See here for Ivar Otto Bendixson). A nice reference is Alekos Kechris‘s book, Classical descriptive set theory. For the Baire category theorem and basic applications, I recommend the beginning of John Oxtoby‘s short book, Measure and category. See also the nice paper Subsum Sets: Intervals, Cantor Sets, and Cantorvals by Zbigniew Nitecki, downloadable at the arXiv.

Homework set 3 (Due March 11). Exercises 3.2.1, 3.2.3, 3.2.7, 3.2.9, 3.2.11, 3.2.12, 3.2.14, 3.3.2, 3.3.4 – 3.3.7, 3.3.9, 3.3.10, 3.4.2, 3.4.4, 3.4.5, 3.4.7 – 3.4.10, 3.5.4 – 3.5.6.

Solution to 3.3.6.

  • March 11 – March 20. Chapter 4. Limits and continuity: “Continuous” limits. Continuity of functions. The interaction of continuity and compactness.  The intermediate value theorem.
  • March 11. The concept of function. Dirichlet‘s and Thomae‘s examples. Definition of limit and basic properties.
  • March 13. Properties of limits (continued). Definition of continuity and basic properties.
  • March 18. Applications of continuity: The intermediate value property. Banach‘s fixed point theorem.
  • March 20.  Continuity and compactness. Uniform continuity. Sets of discontinuity of functions.
  • Additional topics: The history of the concept of function is very interesting. The intermediate value property also has a curious history. Apparently, for a while it was expected that it sufficed to characterize continuity. Bolzano’s original paper is fairly accessible. A particularly interesting continuous function is the Cantor function, also called the devil’s staircase. The topic of fixed points (Exercise 4.5.7) leads to a beautiful theorem of Sharkovski, on the possible periods of continuous functions (See here for Oleksandr Mykolaiovych Sharkovsky).

Homework set 4 (Due April 1st). Exercises 4.2.1, 4.2.4, 4.2.6, 4.2.7, 4.3.1, 4.3.3, 4.3.4, 4.3.6, 4.3.8 – 4.3.10, 4.3.12, 4.4.1, 4.4.4, 4.4.6, 4.4.9, 4.4.10, 4.4.13, 4.5.2, 4.5.4, 4.5.7.

  • April 1 – April 10. Chapter 5. Derivatives: What is a derivative? Differentiability and continuity. Darboux theorem. The mean value theorem. Nowhere differentiable functions.
  • April 1. Sets of discontinuity of functions. Definition of derivative, basic properties. Baire class 1 functions.
  • April 3. Darboux theorem (the intermediate value property).
  • April 8. Rolle‘s theorem. The mean value theorem. L’Hôpital’s rule (see here for Guillaume de l’Hôpital).
  • April 10. Continuous nowhere differentiable functions. Weierstrass function. Proper understanding of this topic requires the notion of uniform convergence, that we will discuss in Chapter 6.
  • Supplemental reading: This is a very useful exercise to review the notions of continuity and uniform continuity. For more on the Baire classes of functions, I recommend Kechris’s book on Classical descriptive set theory. The problem of characterizing which functions are derivatives has led to a significant amount of research; these two notes (by Andrew Bruckner, and by Bruckner and J. L. Leonard) discuss some details: 1, 2. On continuous nowhere differentiable functions, the thesis linked to above (by Johan Thim) is a useful resource. Sections 1, 2, 4 of this “quiz” (by Louis A. Talman) complement well the discussion of similar topics in the book. For the history of the mean value theorem, see these slides by Ádám Besenyei.

Homework set 5 (Due April 15). Exercises 5.2.1 – 5.2.5, 5.3.2, 5.3.3, 5.3.5, 5.3.7.

  • April 15 – April 24. Chapter six: Sequences and series of functions. Pointwise vs. uniform convergence. Uniform convergence, continuity, and differentiability. Power series, Taylor series, C^\infty vs. real analytic.
  • April 15. Pointwise and uniform convergence of sequences of functions. The uniform limit of a sequence of continuous functions is continuous.
  • April 17. Section 6.3: Let (f_n)_{n=1}^\infty be a sequence of differentiable functions defined on a closed interval, that converges pointwise and such that their derivatives converge uniformly. Then the pointwise limit is indeed uniform, the resulting function is differentiable, and its derivative is the limit of the f_n'.
  • April 22. Series of functions. Weierstrass M-test. Power series.
  • April 24. Power series (continued). Taylor series. Real analytic functions.
  • Supplemental reading: On the topic of analytic vs C^\infty functions, see these two essays by Dave L. Renfro: 1, 2. The result of section 6.3 is false if we ask that the sequence of functions f_n converges uniformly while their derivatives converge pointwise. Darji in fact proved that we can have the limit of the f_n be a differentiable function whose derivative disagrees everywhere with the limit of the derivatives. See here. On Formal power series and applications in combinatorics, I recommend the nice paper by Ivan Niven on this topic. For more on real analytic functions, see the first two chapters of the book A primer of real analytic functions, by Steven Krantz and Harold Parks.

Homework set 6 (Due April 29). Exercises 6.2.1, 6.2.5, 6.2.8, 6.2.13, 6.2.15, 6.2.16, 6.3.1, 6.3.4, 6.4.1, 6.4.3 – 6.4.6, 6.5.1, 6.5.2, 6.6.1, 6.6.6.

  • April 29 – May 8. Chapter seven: The Riemann integral. Darboux’s characterization. Basic properties. The fundamental theorem of calculus. Lebesgue‘s criterion.
  • April 29. Darboux’s approach to the Riemann integral in terms of upper and lower sums. Continuous functions are integrable.
  • May 1. Basic properties of the integral, integrable discontinuous functions. A theorem on uniform convergence ensuring that the integral of a limit is the limit of the integrals.
  • May 6. The fundamental theorem of calculus. Sets of measure zero.
  • May 8. Lebesgue’s characterization of Riemann integrable functions.
  • Supplemental reading: For the interesting history of the early development of the Riemann integral, I suggest the first two chapters of Lebesgue’s theory of integration, by Thomas Hawkins.

Homework set 7 (Due May 13 at 10:30). Exercises 7.2.2, 7.2.5, 7.2.6, 7.3.1, 7.3.3, 7.3.6, 7.4.2, 7.4.4, 7.4.6, 7.5.1, 7.5.4, 7.5.10.

Group project due May 15 at 10:30.

A multiple-choice quiz, by Vicky Neale.

 

Grading: Based on homework. There will also be a group project, that will count as much as two homework sets. I expect there will be no exams, but if we see the need, you will be informed reasonably in advance.

There is bi-weekly homework, due Tuesdays at the beginning of lecture; you are welcome to turn in your homework early, but I will not accept homework past Tuesdays at 12:05 pm, or grant extensions. The homework covers some routine and some more challenging exercises related to the topics covered in the past two weeks (roughly, one homework set per chapter). It is a good idea to work daily on the homework problems corresponding to the material covered that day.

You are encouraged to work in groups and to ask for help. However, the work you turn in should be written on your own. Give credit as appropriate: Make sure to list all books, websites, and people you collaborated with or consulted while working on the homework. If relevant, indicate what software packages you used, and include any programs you may have written, or additional data.

Your homework must follow the format developed by the mathematics department at Harvey Mudd College. You will find that format at this link. If you do not use this style, unfortunately your homework will be graded as 0. In particular, please make sure that what you turn in is not your scratch work but the final product. Include partial attempts whenever you do not have a full solution.

I may ask you to meet with me to discuss details of sets, and I suggest that before you turn in your work, you make a copy of it, so you can consult it if needed.

I post links to supplementary material on Google+. Circle me and let me know if you are interested, and I’ll add you to my Analysis circle.

Twitter.

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Riemann on Riemann sums

November 16, 2013

Though Riemann sums had been considered earlier, at least in particular cases (for example, by Cauchy), the general version we consider today was introduced by Riemann, when studying problems related to trigonometric series, in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. This was his Habilitationsschrift, from 1854, published posthumously in 1868.

Riemann’s papers (in German) have been made available by the Electronic Library of Mathematics, see here. The text in question appears in section 4, Ueber den Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit. The translation below is as in

  • A source book in classical analysis. Edited by Garrett Birkhoff. With the assistance of Uta Merzbach. Harvard University Press, Cambridge, Mass., 1973. MR0469612 (57 #9395).

Also zuerst: Was hat man unter \displaystyle \int_a^b f(x) \, dx zu verstehen?

Um dieses festzusetzen, nehmen wir zwischen a und b der Grösse nach auf einander folgend, eine Reihe von Werthen x_1, x_2,\ldots, x_{n-1} an und bezeichnen der Kürze wegen x_1 - a durch \delta_1, x_2 - x_1 durch \delta_2,\ldots, b - x_{n-1} durch \delta_n und durch \varepsilon einen positiven ächten Bruch.  Es wird alsdann der Werth der Summe

\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots \displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)

von der Wahl der Intervalle \delta und der Grössen \varepsilon abhängen.  Hat sie nun die Eigenschaft, wie auch \delta und \varepsilon gewählt werden mögen, sich einer festen Grenze A unendlich zu nähern, sobald sämmtliche \delta unendlich klein werden, so heisst dieser Werth \displaystyle \int_a^b f(x) \, dx.

In Birkhoff’s book:

First of all: What is to be understood by \displaystyle \int_a^b f(x)\,dx?

In order to establish this, we take the sequence of values x_1,x_2,\ldots, x_{n-1} lying between a and b and ordered by size, and, for brevity, denote x_1 - a by \delta_1, x_2 - x_1 by \delta_2,\ldots, b - x_{n-1} by \delta_n, and proper positive fractions by \varepsilon_i. Then the value of the sum

\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots \displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)

will depend on the choice of the intervals \delta_i and the quantities \varepsilon_i. If it has the property that, however the \delta_i and the \varepsilon_i may be chosen, it tends to a fixed limit A as soon as all the \delta_i become infinitely small, then this value is called \displaystyle \int_a^b f(x) \, dx.

(Of  course, in modern presentations, we use \Delta_i instead of \delta_i, and say that the \delta_i approach 0 rather than become infinitely small. In fact, we tend to call the collection of data x_1,\dots,x_{n-1}, \varepsilon_1,\dots,\varepsilon_n a tagged partition of {}[a,b], and call the maximum of the x_{i+1}-x_i the mesh or norm of the partition.)

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515 – Homework 2

February 14, 2012

This set is due Feb. 29 at the beginning of lecture. Let me know if more time is needed or anything like that. Problem 4 was incorrect as stated; I have fixed it now. Thanks to Tara Sheehan for bringing the problem to my attention.

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