MTH322: Real Analysis II (Fall 2021)

This course is offered to MSc, Semester II at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in Real Analysis I.

Sequences of functions: convergence, uniform convergence, uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation, the exponential and logarithmic function, the trigonometric functions.

Series of functions: Absolute convergence, uniform convergence, Cauchy criterion, Weiestrass M-test, power series of functions, radius of convergence, Cauchy-Hadamard theorem, differentiation theorem, uniqueness theorem.

Improper integrals: Improper integral of first and second kind, comparison tests, Cauchy condition for infinite integrals, absolute convergence, absolute convergence of improper integral, uniform convergence of improper integrals, Cauchy condition for uniform convergence, Weiestrass M-test for uniform convergence.

Notes

Sample Question

Chapter 01

  1. Define improper integral of first kind.
  2. Define improper integral of second kind.
  3. Define conveyance of improper integral of first kind.
  4. Consider and integral $\int_{1}^{\infty }{{{x}^{-p}} dx}$, where $p$ is any real number. Discuss its convergence or divergence.
  5. Suppose that $f\in \mathcal{R}[a,b]$ for every $b\ge a$. Assume that $f(x)\ge 0$ for each $x\ge a$. Then prove that $\int_{a}^{\infty }{f(x) dx}$ converges if, and only if, there exists a constant $M>0$ such that $\int\limits_{a}^{b}{f(x)\,dx} \le M$ for every $b\ge a$.
  6. Assume $f\in \mathcal{R}[a,b]$ for every $b\ge a$. If $0\le f(x)\le g(x)$ for every $x\ge a$ and $\int_{a}^{\infty }{g\,dx}$ converges, then $\int_{a}^{\infty }{f\,dx}$ converges and we have $\int_{a}^{\infty }{f\,dx}\le \int_{a}^{\infty }{g\,dx}$.
  7. Suppose that $f,g\in \mathcal{R}[a,b]$ for every $b\ge a$, where $f(x)\ge 0$ and $g(x)\ge 0$ for $x\ge a$. If $\lim\limits_{x\to \infty }\frac{f(x)}{g(x)}=1$, then $\int\limits_{a}^{\infty }{f dx}$ and $\int\limits_{a}^{\infty }{g dx}$ both converge, or both diverge.
  8. Assume that $f\in \mathcal{R}[a,b]$ for every $b\ge a$. Then the integral $\int\limits_{a}^{\infty }{f dx}$ converges if, and only if, for every $\varepsilon >0$ there exists a $B>0$ such that $c>b>B$ implies $\left| \,\int\limits_{b}^{c}{f\,\,dx}\, \right|<\varepsilon $.
  9. Suppose $f\in \mathcal{R}[a,b]$ for every $b\ge a$ and for every $\varepsilon >0\,$there exists a $B>0\,$ such that \[\left| \,\int_{b}^{c}{f\,\,dx}\, \right|<\varepsilon \] for $b,c>B$, then $\int_{a}^{\infty }{f\,dx}$ is convergent.
  10. Define absolutely convergent and conditionally convergent integral.
  11. If $f\in \mathcal{R}[a,b]$ for every $b\ge a$ and if $\int\limits_{a}^{\infty }{f\,dx}$ is absolutely converges, then it is convergent.
  12. If $f(x)$ is bounded for all $x\ge a$, integrable on every closed subinterval of $[a,\infty )$ (i.e. $f\in \mathcal{R}[a,b]$ for each $b\ge a$) and $\int_{a}^{\infty }{g(x)dx}$ is absolutely convergent, then $\int_{a}^{\infty }{f(x)g(x)dx}$ is absolutely convergent.
  13. If $f(x)$ is bounded and monotone for all $x\ge a$ and $\int_{a}^{\infty }{g(x)dx}$ is convergent, then prove that $\int_{a}^{\infty }{f(x)g(x)dx}$ is convergent.
  14. State and prove Abel's theorem for infinite integral.
  15. If $f(x)$ is bounded, monotone for all $x\ge a$ and $\lim\limits_{x\to \infty } f(x)=0$. Also $\int_{a}^{X}{g(x)dx}$ is is bounded for all $X\ge a$, then $\int_{a}^{\infty }{f(x)g(x)dx}$ is convergent.
  16. State and prove Dirichlet theorem for infinite integral.
  17. Use Dirichlet's theorem to prove that $\int\limits_{0}^{\infty }{\frac{\sin x}{x}\,dx}$ is convergent.
  18. Use Dirichlet's theorem to prove that $\int\limits_{1}^{\infty }{\sin {{x}^{2}} dx}$ is convergent.
  19. Use Abel's theorem to prove that $\int\limits_{0}^{\infty }{{e}^{-x} \frac{\sin x}{x} dx}$ is convergent.
  20. Discuss the convergence or divergence of $\int\limits_{0}^{b}{{x}^{-p}} dx$ for real $p$.

Chapter 02

  1. Define pointwise convergence of sequence of function.
  2. Define uniform convergence of sequence of function.
  3. Define pointwise convergence of series of function.
  4. Define uniform convergence of series of function.
  5. State and prove Cauchy’s criterion for uniform convergence of sequence of functions.
  6. State and prove Cauchy’s criterion for uniform convergence of series of functions.
  7. Consider a sequence of function $\{f_n\}$, where $f_n(x)=\frac{nx}{1+n^2 x^2}$, for all $x\in\mathbb{R}$. Prove that $\{f_n\}$ is pointwise convergent but not uniformly convergent on an interval containing $0$.
  8. Let $\{f_n\}$ be a sequence of functions, such that $\lim\limits_{n\to\infty} f_n(x) = f(x)$, $x\in[a,b]$ and let $M_n=\sup_{x\in[a,b]} \left|f_n(x)-f(x) \right|$. Then $f_n\to f$ uniformly on $[a,b]$ if and only if $M_n\to 0$ as $n\to \infty$.
  9. Prove Prove that the sequence $\{f_n\}$, where $f_n(x)=\frac{x}{1+n x^2}$ is uniformly convergent on any interval $I$.
  10. Show that the sequence $\{f_n\}$, where $f_n(x)=nxe^{-nx^2}$, $x\geq 0,$ is not uniformly convergent on $[0,k]$, $k>0$.
  11. Show that the sequence $\{x^n\}$ is not uniformly convergent on $[0,1]$.
  12. Show that the sequence $\{\exp(-nx)\}$ is not uniformly convergent on $[0,k]$, $k>0$.
  13. Test the for uniform convergence of $\displaystyle \left\{\frac{\sin nx}{\sqrt{n}}\right\}$, $0\leq x\leq 2\pi$.
  14. Test the for uniform convergence of $\displaystyle \left\{\frac{x}{n+x}\right\}$, $0\leq x\leq k$, where $k>0$.
  15. Test the for uniform convergence of $\displaystyle \left\{\frac{x}{n+x}\right\}$, $0\leq x< \infty$.
  16. State and prove Weierstrass’s M-test.
  17. Prove that the following series are uniformly convergent for all real $x$: (i) $\sum\frac{\sin (x^2+n^2x)}{n(n+1)}$ (ii) $\sum\frac{(-1)^nx^{2n}}{n^{p+1}(1+x^{2n})}, p>0.$
  18. Let $\{f_n\}$ be a sequence of functions defined on an interval $I$, and $x_0\in I$. If the sequence $\{f_n\}$ converges uniformly to some function $f$ on $I$ and if each of the function $f_n$ is continuous at $x_0$, then the function $f$ is also continuous at $x_0$.
  19. Let $\{f_n\}$ be a sequence of functions defined on an interval $I$. If the sequence $\{f_n\}$ converges uniformly to some function $f$ on $I$ and if each of the function $f_n$ is continuous on $I$, then the function $f$ is also continuous on $I$.
  20. Let $\{f_n\}$ be a sequence of functions defined on $[a,b]$. If $f_n \to f$ uniformly on $[a,b]$ and each function $f_n$ is continuous on $[a,b]$, then $\int_{a}^{b} f(x) dx = \lim\limits_{n\to\infty} \int_{a}^{b} f_n(x) dx.$
  21. Let $\{f_n\}$ be a sequence of functions defined on $[a,b]$ such that ${f_n(x_0)}$ converges for some point $x_0$ on $[a, b]$. If each $f_n$ is differentiable and $\{f'_n\}$ converges uniformly on $[a, b]$, then $\{f_n\}$ converges uniformly on $[a, b]$, to a function $f$, and $f'(x) = \lim_{n\to\infty} f'_n(x), (a < x < b)$.

Chapter 3

  1. \item Consider a sequence of functions $E_n:\mathbb{R}\to\mathbb{R}$ defined as follows: $E_1(x)=1+x$ and $E_{n+1}(x)=1+\int_{0}^{x}E_n(t)dt$ for all $n\in\mathbb{N}$, $x\in\mathbb{R}$. Prove that $E_n$ is well-defined.
  2. Consider a sequence of functions $E_n:\mathbb{R}\to\mathbb{R}$ defined by $E_1(x)=1+x$ and $E_{n+1}(x)=1+\int_{0}^{x}E_n(t)dt$ for all $n\in\mathbb{N}$, $x\in\mathbb{R}$. Prove that for all $n\in \mathbb{N}$, we have $E_n(x)=1+\frac{x}{1!}+\frac{x^2}{2!}+... +\frac{x^n}{n!}$ for all $x\in\mathbb{R}$.
  3. Prove that $\displaystyle \lim_{n\to\infty}\frac{A^n}{n!}=0$ for $A>0$.
  4. Prove that if $\{s_n \}$ is convergent then $\lim\limits_{n\to\infty} s_{n+1}=\lim\limits_{n\to\infty} s_{n}$.
  5. Consider a sequence of function $\{E_n(x)\}$ define by $E_n(x)=1+\frac{x}{1!}+\frac{x^2}{2!}+... +\frac{x^n}{n!}$ for all $x\in\mathbb{R}$. Prove that $\{E_n\}$ converges uniformly on the interval $[-A,A]$, where $A>0$.
  6. Prove that there exists a function $E:\mathbb{R} \to \mathbb{R}$ such that $E'(x)=E(x)$ for all $x \in \mathbb{R}$ and $E(0)=1$.
  7. Define an exponential function.
  8. Define logarithm function.
  9. Define sine function.
  10. Define cosine function.

Assignments and Quizzes

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Videos

Online resources

  1. Brian S. Thomson, Judith B. Bruckner, and Andrew M. Bruckner, Elementary Real Analysis:Second Edition (2008) URL: http://classicalrealanalysis.info/Elementary-Real-Analysis.php
  2. Rudin, W. (1976). Principle of Mathematical Analysis, McGraw Hills Inc.
  3. Bartle, R.G., and D.R. Sherbert, (2011): Introduction to Real Analysis, 4th Edition, John Wiley & Sons, Inc.
  4. Apostol, Tom M. (1974), Mathematical Analysis, Pearson; 2nd edition.
  5. Somasundaram, D., and B. Choudhary, (2005) A First Course in Mathematical Analysis, Narosa Publishing House.
  6. S.C. Malik and S. Arora, Mathematical analysis, New Age International, 1992. (Online google preview)
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