620-295 Real Analysis with applications

Problem Sheet 3

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 23 August 2009

Cardinality

1. Define the following and give an example for each:

   (a)   cardinality,

   (b)   finite,

   (c)   infinite,

   (d)   countable,

   (e)   uncountable.

   
   
2. Show that $\mathrm{Card}\left({\mathbb{Z}}_{>0}\right)=\mathrm{Card}\left({\mathbb{Z}}_{\ge 0}\right)$.
   
   
3. Show that $\mathrm{Card}\left({\mathbb{Z}}_{>0}\right)=\mathrm{Card}\left(\mathbb{Z}\right)$.
   
   
4. Show that $\mathrm{Card}\left({\mathbb{Z}}_{>0}\right)=\mathrm{Card}\left(\mathbb{Q}\right)$.
   
   
5. Show that $\mathrm{Card}\left({\mathbb{Z}}_{>0}\right)\ne \mathrm{Card}\left(ℝ\right)$.
   
   
6. Show that $\mathrm{Card}\left(ℂ\right)=\mathrm{Card}\left(ℝ\right)$.
   
   
7. Let $S$ be a set. Show that $\mathrm{Card}\left(S\right)=\mathrm{Card}\left(S\right)$.
   
   
8. Show that if $\mathrm{Card}\left(S\right)=\mathrm{Card}\left(T\right)$ then $\mathrm{Card}\left(T\right)=\mathrm{Card}\left(S\right)$.
   
   
9. Show that if $\mathrm{Card}\left(S\right)=\mathrm{Card}\left(T\right)$ and $\mathrm{Card}\left(T\right)=\mathrm{Card}\left(U\right)$ then $\mathrm{Card}\left(S\right)=\mathrm{Card}\left(U\right)$.
   
   
10. Define $\mathrm{Card}\left(S\right)\le \mathrm{Card}\left(T\right)$ if there exists an injective function $f:S\to T$. Show that if $\mathrm{Card}\left(S\right)\le \mathrm{Card}\left(T\right)$ and $\mathrm{Card}\left(T\right)\le \mathrm{Card}\left(S\right)$ then $\mathrm{Card}\left(S\right)=\mathrm{Card}\left(T\right)$.
    
    

Sequences

1. Define the following and give an example for each:

   (a)   sequence,

   (b)   converges (for a sequence),

   (c)   diverges (for a sequence),

   (d)   limit (of a sequence),

   (e)   sup (of a sequence),

   (f)   inf (of a sequence),

   (g)   lim sup (of a sequence),

   (h)   lim inf (of a sequence),

   (i)   bounded (for a sequence),

   (j)   increasing (for a sequence),

   (k)   decreasing (for a sequence),

   (l)   monotone (for a sequence),

   (m)   Cauchy sequence,

   (m)   contractive sequence,

   
   
2. Prove that if $\left(a_n\right)$ converges then ${\displaystyle {\lim }_{n\to \infty }{a}_n}$ is unique.
   
   
3. Prove that if $\left(a_n\right)$ converges then $\left(a_n\right)$ is bounded.
   
   
4. Prove that if ${\displaystyle {\lim }_{n\to \infty }{a}_n=a}$ and ${\displaystyle {\lim }_{n\to \infty }{b}_n=b}$ then ${\displaystyle {\lim }_{n\to \infty }{a}_n+b_n=a+b}$.
   
   
5. Prove that if ${\displaystyle {\lim }_{n\to \infty }{a}_n=a}$ and ${\displaystyle {\lim }_{n\to \infty }{b}_n=b}$ then ${\displaystyle {\lim }_{n\to \infty }{a}_n{b}_n=ab}$.
   
   
6. Prove that if ${\displaystyle {\lim }_{n\to \infty }{a}_n=a}$ and ${\displaystyle {\lim }_{n\to \infty }{b}_n=b}$ and $b_n\ne 0$ for all $n\in {\mathbb{Z}}_{>0}$ then ${\displaystyle {\lim }_{n\to \infty }}\frac{a_n}{b_n}=\frac{a}{b}$.
   
   
7. Prove that if ${\displaystyle {\lim }_{n\to \infty }{a}_n=\ell }$ and ${\displaystyle {\lim }_{n\to \infty }{c}_n=\ell }$ and $a_n\le b_n\le c_n$ for all $n\in {\mathbb{Z}}_{>0}$ then ${\displaystyle {\lim }_{n\to \infty }}{b}_n=\ell$.
   
   
8. Prove that if $\left(a_n\right)$ is increasing and bounded above then $\left(a_n\right)$ converges.
   
   
9. Prove that if $\left(a_n\right)$ is increasing and not bounded above then $\left(a_n\right)$ diverges.
   
   
10. Prove that if $\left(a_n\right)$ is decreasing and bounded below then $\left(a_n\right)$ converges.
    
    
11. Prove that if $\left(a_n\right)$ is decreasing and not bounded below then $\left(a_n\right)$ diverges.
    
    
12. Prove that every sequence $\left(a_n\right)$ of real numbers has a monotonic subsequence.
    
    
13. (Bolzano-Weirstrass) Prove that every sequence $\left(a_n\right)$ of real or complex numbers has a convergent subsequence.
    
    
14. Prove that every Cauchy sequence $\left(a_n\right)$ of real or complex numbers converges.
    
    
15. Prove that every convergent sequence $\left(a_n\right)$ is a Cauchy sequence.
    
    
16. Graph and determine the sup, inf, lim sup, lim inf and convergence of the following sequences:

    (a)   $a_n={\left(-1\right)}^n$,

    (b)   ${\displaystyle a_n=\frac{1}{n}}$,

    (c)   ${\displaystyle a_n=\frac{{\left(n\text{!}\right)}^2{5}^n}{\left(2n\right)\text{!}}}$,

    (d)   $a_1=3$, $a_n=\frac{1}{2}{\displaystyle (a_{n-1}+\frac{5}{a_{n-1}})}$,

    (e)   $a_n={(1+\frac{1}{n})}^n$,

    (f)   $a_n=e^{in\pi /7}$,

    
    
17. Does the sequence given by ${\displaystyle \frac{n}{2n+1}}$ converge? If so, what is the limit?
    
    
18. Does the sequence given by $\sqrt{n}$ converge? If so, what is the limit?
    
    
19. Does the sequence given by ${\displaystyle \frac{1}{\sqrt{n}}}$ converge? If so, what is the limit?
    
    
20. Does the sequence given by $\sqrt{n+1}-\sqrt{n}$ converge? If so, what is the limit?
    
    
21. Does the sequence given by $\sqrt{n}(\sqrt{n+1}-\sqrt{n})$ converge? If so, what is the limit?
    
    
22. Does the sequence given by ${\displaystyle \frac{n}{n^2+1}}$ converge? If so, what is the limit?
    
    
23. Does the sequence given by ${\displaystyle \frac{2n}{n+1}}$ converge? If so, what is the limit?
    
    
24. Does the sequence given by ${\displaystyle \frac{3n+1}{2n+5}}$ converge? If so, what is the limit?
    
    
25. Does the sequence given by ${\displaystyle \frac{n^2-1}{2n^2+3}}$ converge? If so, what is the limit?
    
    
26. Show that the sequence $a_n={\displaystyle {(1+\frac{1}{n})}^n}$ is increasing and bounded above by 3.
    
    
27. Let $a\in ℝ$ with $\left|a\right|<1$. Does the sequence given by $a^n$ converge? If so, what is the limit?
    
    
28. Let $a\in ℝ$ with $a>0$. Does the sequence given by $a^{1/n}$ converge? If so, what is the limit?
    
    
29. Does the sequence given by $n^{1/n}$ converge? If so, what is the limit?
    
    
30. Let $a\in ℝ$ with $a>0$. Fix a positive real number $x_1$. Let $x_{n+1}=\frac{1}{2}(x_n+a/x_n)$. Show that the sequence $x_n$ converges to $\sqrt{a}$.
    
    
31. Let $\alpha ,\beta \in {ℝ}_{>0}$. Let $a_1=\alpha$ and $a_{n+1}=\sqrt{\beta +a_n}$. Show that the sequence $a_n$ converges and find the limit.
    
    
32. Let $\alpha ,\beta \in {ℝ}_{>0}$. Let $a_1=\alpha$ and $a_{n+1}=\beta +\sqrt{a_n}$. Show that the sequence $a_n$ converges and find the limit.
    
    
33. Let $x_1=1$ and $x_{n+1}=\frac{1}{2+x_n}$. Show that the sequence $x_n$ converges and find the limit.
    
    
34. Fix a real number $x_1$ between 0 and 1. Let $x_{n+1}=\frac{1}{7}(x_n^3+2)$. Show that the sequence $x_n$ converges and that the limit is a soluntion to the equation $x^3-7x+2=0$. Use this observation to estimate the solution to $x^3-7x+2=0$ to three decimal places.
    
    
35. Find the upper and lower limits of the sequence ${\left(-1\right)}^n(1+\frac{1}{n})$.
    
    
36. Find the upper and lower limits of the sequence given by $a_1=0$, $a_{2k}=\frac{1}{2}{a}_{2k+1}$, and $a_{2k+1}=\frac{1}{2}+a_{2k}$.
    
    
37. Give an example of a sequence $\left(a_n\right)$ such that none of $\inf a_n$, $\lim \inf a_n$, $\lim \sup a_n$, and $\sup a_n$ are equal.
    
    
38. Let $a_n$ be a bounded sequence. Show that $\lim \inf a_n\le \lim \sup a_n$.
    
    
39. Let $a_n$ be a bounded sequence. Show that $a_n$ converges if and only if $\lim \sup a_n\le \lim \inf a_n$.
    
    
40. Let $a_n$ be a bounded sequence such that $\lim \sup a_n\le \lim \inf a_n$. Show that $\lim \sup a_n=\lim \inf a_n=\lim a_n$.
    
    
41. Let $a_n$ be a real sequence. Show that ${\lim }_{n\to \infty }{a}_n=a$ if and only if $\lim \sup a_n=\lim \inf a_n=a$.
    
    
42. Prove that ${\displaystyle \sum _{k=1}^n\frac{1}{k(k+1)}=\frac{n}{n+1}}$.
    
    

Convergence theorems for sequences

1. Prove that a real sequence can have at most one limit.
   
   
2. Prove that every convergent sequence is Cauchy.
   
   
3. Prove that every Cauchy sequence which has a convergent subsequence is itself convergent.
   
   
4. Prove that every Cauchy sequence is bounded.
   
   
5. Prove that every convergent sequence is bounded.
   
   
6. Prove that a contractive sequence is Cauchy.
   
   
7. Prove that a contractive sequence is convergent.
   
   

Series

1. Define the following and give an example for each:

   (a)   series,

   (b)   converges (for a series),

   (c)   diverges (for a series),

   (d)   limit (of a series),

   (e)   absolutely convergent,

   (f)   conditionally convergent,

   (g)   geometric series,

   (h)   harmonic series,

   
   
2. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^5}}$ converges. Use the integral test.
   
   
3. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2+4}}$ converges. Use the integral test.
   
   
4. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^{\mathrm{1/2}}}}$ converges. Use the integral test.
   
   
5. Determine if the series ${\displaystyle \sum _{n=2}^{\infty }\frac{1}{{(n-1)}^2}}$ converges. Use the integral test.
   
   
6. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2+1}}$ converges. Use the comparison test.
   
   
7. Determine if the series ${\displaystyle \sum _{n=2}^{\infty }\frac{n}{n^3-1}}$ converges. Use the comparison test.
   
   
8. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n+1}}$ converges. Use the comparison test.
   
   
9. Determine if the series ${\displaystyle \sum _{n=2}^{\infty }\frac{1}{n-1}}$ converges. Use the comparison test.
   
   
10. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{n}{n^2+1}}$ converges. Use the comparison test.
    
    
11. Determine if the series ${\displaystyle \sum _{n=2}^{\infty }\frac{1}{\sqrt{n}-1}}$ converges. Use the comparison test.
    
    
12. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{2}{3^n+1}}$ converges. Use the comparison test.
    
    
13. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{3^n+1}{4^n+1}}$ converges. Use the comparison test.
    
    
14. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{n^3}{2^n}}$ converges. Use the ratio test.
    
    
15. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{n\text{!}}{n^n}}$ converges. Use the ratio test.
    
    
16. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{2^n}{n+1}}$ converges. Use the ratio test.
    
    
17. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{2^n}{n\text{!}}}$ converges. Use the ratio test.
    
    
18. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{n^{\mathrm{1/3}}}}$ converges.
    
    
19. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\sqrt{\frac{n}{n+1}}}$ converges.
    
    
20. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^7}}$ converges.
    
    
21. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{\sqrt{n^2+n}}}$ converges.
    
    
22. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{n^3}{4^n}}$ converges.
    
    
23. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{\sin n}{1+n^2}}$ converges.
    
    
24. Determine if the series $\frac{2}{1}-\frac{2}{2}+\frac{2}{3}-\frac{2}{4}+\frac{2}{5}-\cdots$ converges.
    
    
25. Determine if the series $-\frac{1}{2}+\frac{2}{3}-\frac{3}{4}+\frac{4}{5}-\frac{5}{6}+\cdots$ converges.
    
    
26. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{\log (n+1)}}$ converges.
    
    
27. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^n\frac{n}{n^2+1}}$ converges.
    
    
28. Determine if the series ${\displaystyle \sum _{n=0}^{\infty }\frac{{\left(-2\right)}^n}{n\text{!}}}$ converges absolutely.
    
    
29. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^n\frac{n}{n^2+1}}$ converges absolutely.
    
    
30. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{\cos n}{n^2}}$ converges absolutely.
    
    
31. Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{\log (n+1)}}$ converges absolutely.
    
    

Power series

1. Write out the first four terms of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n+1}}$.
   
   
2. Write out the first four terms of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n\text{!}}}$.
   
   
3. Write out the first four terms of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{{(x-1)}^n}{n}}$.
   
   
4. Write out the first four terms of the series ${\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{x^n}{n+2}}$.
   
   
5. Find the Taylor expansion of $e^x$ at $x=0$.
   
   
6. Find the Taylor expansion of $\sinh x$ at $x=0$.
   
   
7. Find the Taylor expansion of $\frac{1}{1-x}$ at $x=0$.
   
   
8. Find the Taylor expansion of $e^x$ at $x=2$.
   
   
9. Find the Taylor expansion of $\log x$ at $x=1$.
   
   
10. Find the Taylor expansion of $\frac{1}{x^2}$ at $x=1$.
    
    
11. Prove the identity $e^{ix}=\cos x+i\sin x$.
    
    
12. Prove the identity $e^x=\cosh x+\sinh x$.
    
    
13. Find the radius of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n+1}}$.
    
    
14. Find the radius of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{{(x+1)}^n}{{(n+1)}^2}}$.
    
    
15. Find the radius of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n\text{!}}}$.
    
    
16. Find the radius of convergence of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{{(2x-1)}^n}{\sqrt[3]{n}}}$.
    
    
17. Find the interval of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n+1}}$.
    
    
18. Find the interval of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{{(x+1)}^n}{{(n+1)}^2}}$.
    
    
19. Find the interval of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n\text{!}}}$.
    
    
20. Find the interval of convergence of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{{(2x-1)}^n}{\sqrt[3]{n}}}$.
    
    
21. Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }n{x}^{n-1}}$.
    
    
22. Find the sum of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^{n+1}}{n+1}}$.
    
    
23. Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{x^n}{n}}$.
    
    
24. Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{n}{3^{n-1}}}$.
    
    
25. Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n{2}^{n+1}}}$.
    
    
26. Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }n(n-1){\left(\frac{1}{4}\right)}^n}$.
    
    
27. Find the power series representation of ${\displaystyle \frac{1}{1+2x}}$ and determine its radius of convergence.
    
    
28. Find the power series representation of ${\displaystyle \frac{1}{1+x^2}}$ and determine its radius of convergence.
    
    
29. Find the power series representation of ${\displaystyle \frac{x}{1+x}}$ and determine its radius of convergence.
    
    
30. Find the power series representation of ${\displaystyle \frac{1}{{(1+x)}^2}}$ and determine its radius of convergence.
    
    
31. Find the power series representation of $\arctan x$ and determine its radius of convergence.
    
    
32. Find the power series representation of $\log (2+x)$ and determine its radius of convergence.
    
    
33. Find the power series representation of $\int e^{x^3}dx$.
    
    
34. Find the power series representation of ${\displaystyle \int \frac{\sinh x}{x}dx}$.
    
    
35. Find an infinite series representation of ${\displaystyle {\int }_{-1}^1\frac{\sinh x}{x}dx}$.
    
    
36. Find an infinite series representation of ${\displaystyle {\int }_0^1{e}^{x^3}dx}$.