Advanced Calculus II as of

1. Homework I on Sequences and Series: Due Tue, Jan 21 in class

   
   For this set of questions, you should try all the questions 1-4. But turn in only one well written solution of your choice. (Be sure to include "all" the details!). (In my opinion, the difficulty of the questions, from easiest to hardest, is 3,1,2a, 2b,4. But, like most of math, this is a subjective judgement)

   1. If $(\vec x_n)$ is a sequence in $\mathbf{R}^p$ such that $\vec x_n\to \vec a$, show that the sequence of arithmetic means converges to the same limit, $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n \vec x_k =\vec a$. Hint: WLOG $\vec a=\vec 0$.
   2. It is known that the root test is stronger (=better) than the ratio test. To confirm this, prove one of the following claims:
      • If $(a_n)$ is a bounded sequence of strictly positive real numbers, show that $\limsup_{n\to\infty}\sqrt[n]{a_n} \leq \limsup_{n\to\infty} \frac{a_{n+1}}{a_n}$. (Bartle Exercise 18.F:).
      • Alternative version: If $(a_n)$ is a bounded sequence of strictly positive real numbers such that $r=\lim_{n\to\infty} \frac{a_{n+1}}{a_n}$ exists, then $\lim_{n\to\infty}\sqrt[n]{a_n}=r$. .
   3. The Cauchy Condensation Test (Barlet Exercise 34.K, or Rudin Theorem 3.27) is a theorem that says that if a sequence $(a_n)$ of real numbers is monotone decreasing and positive, then series $\sum_{n=1}^\infty a_n$ converges iff series $\sum_{k=0}^\infty 2^k a_{2^k}$ converges.

      Use The Cauchy Condensation Test to prove both of the following statements:

      • For $p\in\mathbf{R}$ the series $\sum_n\frac{1}{n^p}$ converges iff $p>1$.
      • For $p\in\mathbf{R}$ the series $\sum_{n=2}^\infty\frac{1}{n (\log n)^p}$ converges iff $p>1$.
   4. Bartle Exercise 34.K has a hint in the back of the book. Without looking into Rudin's book, use this hint to write the proof of The Cauchy Condensation Test.
2. Homework II on integration Due Tue, Jan 28 in class
   1. Compute one of the following integrals: $\int_{-1}^1 x d|x|$ (with respect to absolute value function), $\int_{0}^2 x^3 d\lfloor x\rfloor$ (with respect to the greatest integer/floor function as in Bartle Example 29.9(e)).
      Hint: It is easier to use properties than the definition of the RS-integral.
   2. If $f $ is positive and continuous on $[0,1]$ and $M=\sup_{x\in[0,1]}f(x)$, show that $\lim_{n\to\infty}\sqrt[n]{ \int_0^1 (f(x))^n dx}=M$.
      Hint: One can use "the squeeze theorem" from calculus or from Bartle Thm 14.9. To put it differently, inequality $\limsup_{n\to\infty}\sqrt[n]{ \int_0^1 (f(x))^n dx}\leq M$ is obvious/easy. Inequality $\liminf_{n\to\infty}\sqrt[n]{ \int_0^1 (f(x))^n dx}\geq M$ may require a lemma which is somewhere in Bartle, but it maybe quicker to prove it than to find it.
   3. Prove that $\lim_{n\to\infty}\int_0^1 \frac{sin (nx)}{nx} dx=0$. (Compare Bartle 31.D). Alternatively, you may choose one of the problems 31.C, 31.E, 31F from Bartle.
3. Homework III on integration Due Tue, Feb 4 in class

   (To be posted)

• Instructor: Wlodek Bryc
  Office French West 4316
  E-mail:
  Office Hours
  Phone: 513-556-4098
• Main Textbook: R. Bartle, The Elements of Real Analysis, 2nd edition. Secondary: W. Rudin, Principles of Mathematical Analysis, 3rd edition.

Exams and important dates. See also UC Academic Calendars

• 2h Exam 1: TBA (target date is around Feb 12). Room 115 W. Charlton
• 2h Exam 2: TBA (target date is around March 26). Room 115 W. Charlton
• Final Exam: Thursday, May 1, 12:30 - 2:30 pm, Room 115 W. Charlton

Course Grade Calculation

Topics

According to course description, material covered includes: linear maps, differentiability, partial derivatives, differentiability of functions whose partial derivatives are continuous, chain rule, Jacobian, inverse and implicit function theorems. Uniform convergence of sequences of functions, Arzela-Ascoli theorem. Basics of Fourier series. Hower, wee will start with some topics from the previous semester.

1. Sequences and series

   • Sequences $(x_n)$ in metric spaces: Dr. Speight notes Ch 4, Bartle Ch III (pg 90), Rudin Ch 3 and Ch 7
   • Numerical Series $\sum_n a_n$: root test, Dirichlet test
     Dr. Speight notes Ch 4, Bartle Ch VI, Rudin Ch 3 and Ch 7
   • Sequences and series of functions $f_n:\mathbf{R}^p\to\mathbf{R}^q$. Uniform convergence, completeness of $C[0,1]$. Bartle Sect 26, Sect. 37, Rudin Ch 7, Dr. Speight notes end of Ch 5.
2. Leftover integration topics: Dr. Speight notes Ch 7 pages 41 - 70. Bartle Sect. 30+31
   1. Mean value theorems
   2. $\lim_{n\to\infty}\int_a^b f_n dg$

      Position marker: We are here, more or less

   3. interchange theorem
   4. Improper integrals, including comparison tests and Dirichlet's Test (Bartle Thm 32.9), "integral test for series" (Bartle Thm. 35.12), and exercises 32.F, 32.G.
3. Some topics that were omitted in Adv. Calc I:

   • The Stone-Weierstrass Theorem (Rudin Thm 7.26, Bartle Thm 24.7/24.8. Do we want Bartle, Thm 26.2? No! $C[0,1]$ is a complete and separable metric space!
   • Equicontinuity and The Arzela-Ascoli Theorem. Bartle Thm 26.7 Rudin Ch 7, Thm 7.25
   • Power series Bartle Sect. 37, Rudin Ch. 8, Dr. Speight notes Ch 6 pgs $\geq 22$
     • Radius of convergence, term-wise integration, term-wise differentiation
     • Multiplication of series
     • Taylor's Thm Bartle 28.6.
4. Exam 1 will be over the material that we manage to cover. You should know how to do problems similar to homework as well as proofs of selected theorems.

   Proofs required in 2024 - to be updated for 2025:

   1. Uniform convergence of continuous functions: Bartle, Theorem 24.1
   2. Absolute convergence of series of vectors: Bartle Thm 34.7
   3. Root test for absolute convergence of numerical series: Bartle Thm 35.3+Corollary 35.4
   4. Dirichlet's test for series: Bartle, Thm 36.2
   5. Radius of convergence for power series: Bartle, Thm 37.13 (Cauchy-Hadamard)
   6. Uniform convergence: Bartle, Thm 37.14
   7. Term by term Differentiation: Bartle, Thm 37.16
   8. Uniqueness: Bartle, Thm 37.17
   9. Improper integrals:
      • Thm 32.5 Cauchy Criterion
      • Thm 32.7 Comparison Test
      • Thm 32.8 Limit Comparison Test
5. Derivatives for functions $\mathbb{R}^p\supset E \to \mathbb{R}^q$.

   • Linear functions (and linear functionals): Bartle Section 21, Rudin Ch 9, Dr. Speith notes Ch 5 pg 20-24
   • The derivative in $\mathbb{R}^p$: Bartle Section 39, Rudin Ch9
   • The chain rule and MVT: Bartle Section 40, Rudin Ch 9
   • Mapping Theorems, Inversion theorem, Implicit functions: Bartle Section 41, Rudin Ch 9
   • Extrema: Bartle Section 42

   Orthogonal expansions - a prequel to Fourier series.
6. Exam 2

   The exam will be over the material that we manage to cover. You should know how to do problems similar to homework as well as proofs of selected theorems.
   Proofs required in 2024 - to be updated for 2025:

   1. Banach fixed point theorem (Bartle Thm 23.5 )
   2. Continuity theorem (Bartle, Lemma 39.5)
   3. Partial derivatives theorem (Bartle Thm 39.6 and Corollary 39.7)
   4. $C^1$ Differentiability criterion (Bartle, Thm 39.9)
   5. Chain Rule (Bartle Thm 40.2)
   6. EZ Mean Value Thm (Bartle Thm 40.4)
   7. First derivative test : Bartle Thm 42.1
   8. Second derivative test: Bartle Thm 42.5
7. Fourier series: Bartle Section 38.
8. Additional topics.

   To be decided later. Currently under consideration:

   • A sequel to Fourier series - Intro to orthogonal polynomials.
   • Problems from past Advanced Calculus Qualifying Exams.
9. Final Exam: Thursday, May 1, 12:30- 2:30 pm, Room 115 W. Charlton

   The final exam will be cumulative over all the material covered. You should know how to do problems similar to homework as well as proofs of selected theorems. (4 questions)
   Required proofs:

   1. List of proofs from Exam 1
      1. Uniform convergence of continuous functions: Bartle, Theorem 24.1
      2. Absolute convergence of series of vectors: Bartle Thm 34.7
      3. Root test for absolute convergence of numerical series: Bartle Thm 35.3+Corollary 35.4
      4. Dirichlet's test for series: Bartle, Thm 36.2
      5. Radius of convergence for power series: Bartle, Thm 37.13 (Cauchy-Hadamard)
      6. Uniform convergence: Bartle, Thm 37.14
      7. Term by term Differentiation: Bartle, Thm 37.16
      8. Uniqueness: Bartle, Thm 37.17
      9. Improper integrals:
         • Thm 32.5 Cauchy Criterion
         • Thm 32.7 Comparison Test
         • Thm 32.8 Limit Comparison Test
   2. List of proofs from Exam 2
      1. Banach fixed point theorem (Bartle Thm 23.5 )
      2. Continuity theorem (Bartle, Lemma 39.5)
      3. Partial derivatives theorem (Bartle Thm 39.6 and Corollary 39.7)
      4. $C^1$ Differentiability criterion (Bartle, Thm 39.9)
      5. Chain Rule (Bartle Thm 40.2)
      6. EZ Mean Value Thm (Bartle Thm 40.4)
      7. First derivative test : Bartle Thm 42.1
      8. Second derivative test: Bartle Thm 42.5
   3. Fourier Series
      • Bartle Lemma 38.3
      • Bartle Theorem 38.4 (Bessel Inequality)

• Jan 14:

  • Theorems, Lemmas, Definitions, Propositions, Corollaries.
    • Lemma $|\sin x|\leq x$. Proof 1: picture Proof 2: calculus. (Can a calculus proof be incorrect?)
    • Proposition If $f$ is odd then $\int_{-1}^1 f(x) dx=0$. Proof: picture (Are picture "proofs" correct? Convincing?)
    • Definition (of what?) $\exists_{L\in\mathbb{R}} \forall_{\epsilon>0} \exists_{ N\geq 1} \forall_{n\in\mathbb{N}, n>N } |a_n-L|<\epsilon$
    • Theorem For sequences of real numbers, the following are equivalen:
      $(a_n)$ converges
      • $(a_n)$ is a Cauchy sequence
      Definition $(a_n)$ converges, if $\forall_{\epsilon>0} \exists_{M}\forall_{n,m>M} |a_n-a_m|<\epsilon$. (Is this an "incorrect" definition?)
  • Geometric progression: Definition + two formulas
    • If $b_{n+1}=r b_n$ then $b_n=b_0 r^n$. (Proof?)
    • If $b_n=b_0 r^n$ and $r\ne 1$ then $b_0+b_1+\dots+b_n=b_0 \frac{1-r^{n+1}}{1-r}$. (Proof?)
  • Binomial law behind a proof of Theorem 24.7 in Bartle. If $p_k(n):=P(X=k)=\begin{pmatrix}n \\ k \end{pmatrix} p^k q^{n-k}$ for $k=0,\dots ,n$ then
    • $\sum_k p_k(n)=1$
    • $E(X):=\sum_{k} k p_k(n)= n p$
    • $Var(X)=\sum_{k} (k-np)^2 p_k(n)= n p q$
    • Chebyshev inequality $P(|X/n-p|>\epsilon)<\frac{p(1-p)}{n \epsilon^2}$ for $\epsilon>0$ imples the law of large numbers: $\forall \epsilon>0$ $\lim_{n\to\infty}P(|X/n-p|>\epsilon)=0$
  • Norms and metrics: Convergent sequences in $\mathbb{R}^p$, $C[0,1]$, $\ell_2$, $\ell_1$, $c$.

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    Norms etc are in exercises/projects for Section 8 in Bartle. Limits in $\mathbb{R}^p$ are in exercises/projects for Secton 15. Series are in exercises for Section 34. Chose an exercise that you find interesting/puzzling and be ready to present it in class.

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  Separable complete metric spaces are the nicest spaces to work with. (What is "metric space"? What is "separable"? What is "complete"?)

  • Consider $C[0,1]$, the space of all continuous functions on $[0,1]$ with the norm $\|f\|_\infty=\max_{x\in[0,1]}|f(x)$. Can we prove that $C[0,1]$ with metric $d(f,g)=\|f-g\|_\infty$ is a separable complete metric space? [Yes, we can!]
  • Consider $C[0,1]$, the space of all continuous functions on $[0,1]$ with the norm $\|f\|_1=\int_0^1|f(x)|dx$. Can we prove that $C[0,1]$ with metric $d_1(f,g)=\|f-g\|_1$ is a separable complete metric space? [No, we cannot prove completeness] Hint: One can find a sequence $(f_n)$ that is Cauchy in metric $d_1$ but does not converge to a continuous function.

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• Jan 23:

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  We may want to discuss the Homework questions

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  Exercise 24.H in Bartle Dini's Theorem Let $(f_n)$ be a monotone decreasing sequence of continuous functions on $[-1,1]$. (That is $f_1(x)\geq f_2(x)\geq \dots$ for all $x\in[-1,1]$.) If $f_n$ converges pointwise to a function $f$ that is continuous on $[-1,1]$ then convergence is uniform. Hint: WLOG $f\equiv 0$.

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  One can find examples that illustrate the importancve of each of the assumptions in Dini's theorem: (I think one can cook up piecewise linear functions for each case)
  • Give an example of a monotone decreasing sequence $(f_n)$ of continuous functions on $[-1,1]$ such that $f_n$ converges pointwise to a function $f$ that is not continuous on $[-1,1]$.
  • Give an example of a sequence $(f_n)$ of continuous functions on $[-1,1]$ such that $f_n$ converges pointwise to a function $f$ that is continuous on $[-1,1]$ but the convergence is not uniform.
  • Give an example of a monotone decreasing sequence $(f_n)$ of continuous functions on $(-1,1)$ such that $f_n$ converges pointwise to a function $f$ that is continuous on $(-1,1)$ but the convergence is not uniform.

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  Binomial law behind a proof of Theorem 24.7 in Bartle. If $p_k(n):=P(X=k)=\begin{pmatrix}n \\ k \end{pmatrix} p^k q^{n-k}$ for $k=0,\dots ,n$ then
  • $\sum_k p_k(n)=1$
  • $E(X):=\sum_{k} k p_k(n)= n p$
  • $Var(X)=\sum_{k} (k-np)^2 p_k(n)= n p q$
  • Chebyshev inequality $P(|X/n-p|>\epsilon)<\frac{p(1-p)}{n \epsilon^2}$ for $\epsilon>0$ imples the law of large numbers: $\forall \epsilon>0$ $\lim_{n\to\infty}P(|X/n-p|>\epsilon)=0$

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  • If you never saw a proof of Holder's inequality perhaps we should go over some of Bartle's Project 8$\beta$.
    Given $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1$, for continuous $f,g$ we have $$\left|\int_0^1 f(x)g(x) dx\right|\le \sqrt[p]{\int_0^1 |f(x)|^p} \sqrt[q]{\int_0^1 |f(x)|^q}$$ Given sequences $(a_n)$ and $(b_n)$ of real numbers such that $\sum_n |a_n|^p<\infty$ and $\sum_n |b_n|^q<\infty$, the series $\sum_na_nb_n$ converges and $$\left|\sum_{n=1}^\infty a_n b_n\right|\leq \sqrt[p]{\sum_n |a_n|^p}\sqrt[q]{\sum_n |b_n|^q}$$ The hard part is to prove inequality in Bartle's Project 8$\beta$(a). But we can just accept "an elementary inequality" $a b\leq \frac{a^p}{p}+\frac{b^q}{q}$ for $a,b>0$ without proof, and use it.
    Note: The inequalities hold also for $p=1,q=\infty$. The best known and most important case is of course $p=q=2$.

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  • Exercises for Euler's formula $e^{i x}=\cos x + i \sin x$ and geometric sum that we will need for Fourier series: There are "nice" formulas for $\sum_{k=1}^n \sin kx$ and $\sum_{k=1}^n \cos k x$ which give an identity $$\tfrac12+\sum_{k=1}^n \cos k x=\frac{\sin (n+\tfrac12)x}{2 \sin \tfrac12 x}$$
  • Another geometric sum exercise: If $A$ is a square matrix with no eigenvalue 1, then $I+A+\dots+A^n = ??$

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  • Jan 30: (Thursday) (new date!)

    To be posted

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  • Feb 6: (Thursday) (new date!)

    To be posted

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