Problem Set: Sequences
620-295 Semester I 2010

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 15 March 2010

(1) Sequence analysis
(2) Definitions and proofs
(3) Picard and Newton iteration

Sequence analysis

For each of the following sequences:

(a) explicitly write out the first 7 terms,

(b) graph the sequence,

(c) determine if it is bounded,

(d) determine if it is increasing or decreasing,

(e) determine if it is Cauchy,

(f) determine the sup and inf,

(g) determine the lim sup and lim inf,

(h) determine if it is convergent or divergent,

(i) determine the limit if it is convergent,

(m) determine if it is contractive.

	Analyse the sequence $a_n=n$.
	Analyse the sequence $a_n={(-1)}^{n}n$.
	Analyse the sequence $a_n=n^2$.
	Analyse the sequence $a_n=12n-n^3$.
	Analyse the sequence $a_n=n\text{!}$.
	Analyse the sequence ${\displaystyle a_n=\frac{1}{n}}$.
	Analyse the sequence ${\displaystyle a_n=3-\frac{1}{n}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{1}{n^p}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{1}{n\text{!}}}$.
	Analyse the sequence $a_n={\displaystyle \frac{n}{n(n+1)}}$.
	Analyse the sequence $a_n={\displaystyle \frac{1}{n}-\frac{1}{n+1}}$.
	Analyse the sequence $a_n={\displaystyle \frac{(-1)}{n+1}}$.
	Analyse the sequence $a_n={\displaystyle \frac{{(-1)}^{n+1}}{n}}$.
	Analyse the sequence $a_n={\displaystyle {(-1)}^n(1+\frac{1}{n})}$.
	Analyse the sequence $a_n={\displaystyle \frac{n}{2n+1}}$.
	Analyse the sequence $a_n={\displaystyle \frac{2n}{n+1}}$.
	Analyse the sequence $a_n={\displaystyle \frac{n}{n^2+1}}$.
	Analyse the sequence $a_n={\displaystyle \frac{3n+1}{2n+5}}$.
	Analyse the sequence $a_n={\displaystyle \frac{n^2-1}{2n^2+3}}$.
	Analyse the sequence $a_n={\displaystyle \frac{i^n}{n^2}}$.
	Analyse the sequence $a_n={\displaystyle \frac{n+2i}{n}}$.
	Analyse the sequence $a_n={\displaystyle \frac{4n+3}{4n^2+3n+1}}$.
	Analyse the sequence $a_k={\displaystyle \frac{1}{{(3k^4-7k^2+5)}^{\frac{1}{3}}}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{{\left(n\text{!}\right)}^2}{\left(2n\right)\text{!}}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{{\left(n\text{!}\right)}^2{5}^n}{\left(2n\right)\text{!}}}$.
	Analyse the sequence $a_n={(-1)}^n$.
	Analyse the sequence $a_n=n^{1/n}$.
	Analyse the sequence $a_n={\displaystyle {(1+\frac{1}{n})}^n}$.
	Analyse the sequence $a_n=e^{in\pi /7}$.
	Analyse the sequence $a_n=\sqrt{n}$.
	Analyse the sequence $a_n={\displaystyle \frac{1}{\sqrt{n}}}$.
	Analyse the sequence $a_n=\sqrt{n+1}-\sqrt{n}$.
	Analyse the sequence $a_n=\sqrt{n}(\sqrt{n+1}-\sqrt{n})$.
	Let $x\in ℝ$ with $\left|x\right|<1$. Analyse the sequence $a_n=x^n$.
	Let $x\in ℝ$ with $x>0$. Analyse the sequence $a_n=x^{1/n}$.
	Let $x\in ℝ$. Analyse the sequence $a_n={\displaystyle {(1+\frac{x}{n})}^n}$.
	Let $x\in ℝ$. Analyse the sequence $a_n={\displaystyle \frac{1-x^{n+1}}{1-x}}$.
	Let $x\in ℝ$. Analyse the sequence $a_n={\displaystyle 1+x+\cdots +x^n}$.
	Analyse the sequence given by $a_1=3$ and $a_n={\displaystyle \frac{1}{2}(a_{n-1}+\frac{5}{a_{n-1}})}$.
	Let $a\in ℝ$ with $a>0$. Fix a positive real number $x_1$. Analyse the sequence given by $x_{n+1}={\displaystyle \frac{1}{2}(x_n+\frac{a}{x_n})}$.
	Let $\alpha ,\beta \in {ℝ}_{>0}$. Analyse the sequence given by $a_1=\alpha$ and $a_{n+1}=\sqrt{\beta +a_n}$.
	Let $\alpha ,\beta \in {ℝ}_{>0}$. Analyse the sequence given by $a_1=\alpha$ and $a_{n+1}=\beta +\sqrt{a_n}$.
	Analyse the sequence given by $x_1=1$ and $x_{n+1}={\displaystyle \frac{1}{2+x_n}}$.
	Fix a real number $x_1$ between 0 and 1. Analyse the sequence given by $x_{n+1}={\displaystyle \frac{1}{7}(x_n^3+2)}$. Estimate the solution to $x^3-7x+2=0$ to three decimal places and verify that the limit is a solution to the equation $x^3-7x+2=0$.
	Analyse the sequence given by $a_1=0$,   $a_{2k}=\frac{1}{2}{a}_{2k+1}$,   and $a_{2k+1}=\frac{1}{2}+a_{2k}$.
	Analyse the sequence ${\displaystyle a_n=\frac{n}{2n+3}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{n}{n+1}-\frac{n+1}{n}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{1-n}{n^3}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{3n-1}{2n+5}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{n+1}{\sqrt{n}}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{\sqrt{n}}{n+1}}$.
	Analyse the sequence ${\displaystyle a_n=1+{(-1)}^{n+1}}$.
	Analyse the sequence ${\displaystyle a_n=n^{{(-1)}^n}}$.
	Analyse the sequence ${\displaystyle a_n=a_n}$ when $a_n=\sqrt{3}$.
	Analyse the sequence ${\displaystyle a_n=\frac{n}{2^n}}$.
	Analyse the sequence ${\displaystyle a_n=\cos \frac{n\pi }{2}}$.
	Analyse the sequence ${\displaystyle a_n=(1+{(-1)}^n)\frac{1}{n}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{4n^2-2n+\cos n}{3n^2+7n+6}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{3n-5}{\sqrt{2n^2+1}+\sqrt{3n^2+4}}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{2n^2+6n+2}{3n^3-n^2-n}}$.
	Analyse the sequence ${\displaystyle a_n=\sqrt{n^2+3n}-\sqrt{n^2+4}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{1}{3n+5}}$.
	Analyse the sequence ${\displaystyle a_n=3+\frac{{(-1)}^n}{n}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{n-2}{n+2}}$.
	Analyse the sequence ${\displaystyle a_n={\left(\frac{n}{25}\right)}^{\mathrm{1/3}}}$.
	Analyse the sequence ${\displaystyle a_n=\frac{2^{2n}}{n\text{!}}}$.
	Analyse the sequence given by $a_1=\sqrt{2}$ and ${\displaystyle a_{n+1}=\sqrt{2+a_n}}$. In particular, show that $a_n$ is increasing and bounded above by 3.
	Analyse the sequence given by $a_1=2$ and ${\displaystyle a_{n+1}=3-\frac{1}{n}}$. In particular, show that $a_n$ is increasing and $a_n<3$ for all $n$.
	Suppose the nth pass through a manufacturing process is modelled by the linear equations $x_n=A^n{x}_0$, where $x_0$ is the initial state of the system and $A=\frac{1}{5}\left(\begin{array}{cc}3& 2\\ 2& 3\end{array}\right)$. Show that $$A^n=\left(\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{2}\end{array}\right)+{\left(\frac{1}{5}\right)}^n\hspace{0.17em}\left(\begin{array}{cc}\frac{1}{2}& -\frac{1}{2}\\ -\frac{1}{2}& \frac{1}{2}\end{array}\right)\text{.}$$ Then with the initial state $x_0=\left(\begin{array}{c}p\\ 1-p\end{array}\right)$, calculate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{x}_n}$.
	The Fibbonaci sequence 0,1,1,2,3,5,8,13, ... is described by the difference equation $F_{k+2}=F_{k+1}+F_k$ and the initial conditions $F_0=0$, $F_1=1$. Writing $u_k=\left(\begin{array}{c}{F}_{k+1}\\ F_k\end{array}\right)$, show that $u_{k+1}=A{u}_k$, where $$A=\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right)\text{.}$$ Solve for $u_k$ in terms of $u_0=\left(\begin{array}{c}1\\ 0\end{array}\right)$ and show that $$F_k=\frac{1}{\sqrt{5}}({\left(\frac{1+\sqrt{5}}{2}\right)}^k-{\left(\frac{1-\sqrt{5}}{2}\right)}^k)\text{,}$$ and therefore find the limit as $k\to \mathrm{\infty }$ of the ratio $F_{k+1}/F_k$.

Definitions and proofs

	What is a sequence?
	What is a convergent sequence?
	What is a divergent sequence?
	What is the limit of a sequence?
	What is the sup of a sequence?
	What is the inf of a sequence?
	What is the lim sup of a sequence?
	What is the lim inf of a sequence?
	What is a bounded sequence?
	What is an increasing sequence?
	What is a decreasing sequence?
	What is a monotone sequence?
	What is a Cauchy sequence?
	What is a contractive sequence?
	Prove that if $\left(a_n\right)$ is a sequence in $ℂ$ and $\left(a_n\right)$ converges then ${\displaystyle {\lim }_{n\to \infty }{a}_n}$ is unique.
	Prove that if $\left(a_n\right)$ is a sequence in $ℂ$ and $\left(a_n\right)$ converges then $\left(a_n\right)$ is bounded.
	Prove that if $\left(a_n\right)$ and $\left(b_n\right)$ are sequences in $ℂ$ and ${\displaystyle {\lim }_{n\to \infty }{a}_n=a}$ and ${\displaystyle {\lim }_{n\to \infty }{b}_n=b}$ then ${\displaystyle {\lim }_{n\to \infty }{a}_n+b_n=a+b}$.
	Prove that if $\left(a_n\right)$ and $\left(b_n\right)$ are sequences in $ℂ$ and ${\displaystyle {\lim }_{n\to \infty }{a}_n=a}$ and ${\displaystyle {\lim }_{n\to \infty }{b}_n=b}$ then ${\displaystyle {\lim }_{n\to \infty }{a}_n{b}_n=ab}$.
	Prove that if $\left(a_n\right)$ and $\left(b_n\right)$ are sequences in $ℂ$ and ${\displaystyle {\lim }_{n\to \infty }{a}_n=a}$ and ${\displaystyle {\lim }_{n\to \infty }{b}_n=b}$ and $b_n\ne 0$ for all $n\in {\mathbb{Z}}_{>0}$ then ${\displaystyle {\lim }_{n\to \infty }}\frac{a_n}{b_n}=\frac{a}{b}$.
	Prove that if $\left(a_n\right)$, $\left(b_n\right)$ and $\left(c_n\right)$ are sequences in $ℝ$ and ${\displaystyle {\lim }_{n\to \infty }{a}_n=\ell }$ 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 ${\displaystyle {\lim }_{n\to \infty }}{b}_n=\ell$.
	Prove that if $\left(a_n\right)$, is a sequence in $ℝ$ and $\left(a_n\right)$ is increasing and bounded above then $\left(a_n\right)$ converges.
	Prove that if $\left(a_n\right)$, is a sequence in $ℝ$ and $\left(a_n\right)$ is not bounded then $\left(a_n\right)$ diverges.
	Find a sequence $a_n$ in $ℝ$ such that if $r\in [0,1]$ there is a subsequence of $a_n$ which converges to $r$.

Picard and Newton iteration

	Let $f:(0,\frac{1}{2}\pi )\to ℝ$ is given by $f\left(x\right)=\frac{1}{2}\tan x$. Estimate numerically the solution to $x=f\left(x\right)$ with $x\in (0,\frac{1}{2}\pi )$ using Picard iteration.
	Let $f:(0,\frac{1}{2}\pi )\to ℝ$ is given by $f\left(x\right)=\frac{1}{2}\tan x$. Estimate numerically the solution to $x=f\left(x\right)$ with $x\in (0,\frac{1}{2}\pi )$ using Newton iteration (let $F\left(x\right)=x-f\left(x\right)$).
	Show that the equation $g\left(x\right)=x^3+x-1=0$ has a solution between 0 and 1. Transform the equation to the form $x=f\left(x\right)$ for a suitable function $f:[0,1]\to [0,1]$. Use Picard iteration to find the solution to 3 decimal places. (Try $f\left(x\right)=1/(x^2+1)$).
	Show that the equation $g\left(x\right)=x^4-4x^2-x+4=0$ has a solution between $\sqrt{3}$ and 2. Transform the equation to the form $x=f\left(x\right)$ for a suitable function $f:[\sqrt{3},2]\to [\sqrt{3},2]$. Use Picard iteration to find the solution to 3 decimal places. (Try $f\left(x\right)=\sqrt{2+\sqrt{x}}$).
	Applying Newton's method to solve the equation $f\left(x\right)=x^2-2=0$ gives a sequence $x_n$ defined recursively by ${\displaystyle x_{n+1}=x_n-\frac{f\left(x_n\right)}{f\prime \left(x_n\right)}}$, for each $n\ge 1$. We choose $x_1=2$ as our initial approximate solution. (a) Verify that ${\displaystyle x_{n+1}=\frac{x_n}{1}+\frac{1}{x_n}}$. Use this to calculate $x_2$ and $x_3$. (b) Show that if the limit ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{x}_n=L}$ exists, then it must satisfy $L^2-2=0$. (c) Show, by induction on $n$, that $\sqrt{2}<x_n\le 2$, for all $n$. (d) Show that $x_{n+1}<x_n$, for all $n$. (e) Deduce that the sequence $x_n$ has a limit, and that ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{x}_n=\sqrt{2}}$.

References

[Ca] S. Carnie, 620-143 Applied Mathematics, Course materials, 2006 and 2007.

[Ho] C. Hodgson, 620-194 Mathematics B and 620-211 Mathematics 2 Notes, Semester 1, 2005.

[Wi] P. Wightwick, UMEP notes, 2010.