Problem Set - Sequences

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: 7 December 2009

Sequences

	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,
	Prove that if $\left(a_n\right)$ converges then ${\displaystyle {\lim }_{n\to \infty }{a}_n}$ is unique.
	Prove that if $\left(a_n\right)$ converges then $\left(a_n\right)$ is bounded.
	Prove that if ${\displaystyle {\lim }_{n\to \infty }{a}_n=a}$ and ${\displaystyle {\lim }_{n\to \infty }{b}_n=b}$ then ${\displaystyle {\lim }_{n\to \infty }{a}_n+b_n=a+b}$.
	Prove that if ${\displaystyle {\lim }_{n\to \infty }{a}_n=a}$ and ${\displaystyle {\lim }_{n\to \infty }{b}_n=b}$ then ${\displaystyle {\lim }_{n\to \infty }{a}_n{b}_n=ab}$.
	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 ${\displaystyle {\lim }_{n\to \infty }}\frac{a_n}{b_n}=\frac{a}{b}$.
	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 ${\displaystyle {\lim }_{n\to \infty }}{b}_n=\ell$.
	Prove that if $\left(a_n\right)$ is increasing and bounded above then $\left(a_n\right)$ converges.
	Prove that if $\left(a_n\right)$ is increasing and not bounded above then $\left(a_n\right)$ diverges.
	Prove that if $\left(a_n\right)$ is decreasing and bounded below then $\left(a_n\right)$ converges.
	Prove that if $\left(a_n\right)$ is decreasing and not bounded below then $\left(a_n\right)$ diverges.
	Prove that every sequence $\left(a_n\right)$ of real numbers has a monotonic subsequence.
	(Bolzano-Weirstrass) Prove that every sequence $\left(a_n\right)$ of real or complex numbers has a convergent subsequence.
	Prove that every Cauchy sequence $\left(a_n\right)$ of real or complex numbers converges.
	Prove that every convergent sequence $\left(a_n\right)$ is a Cauchy sequence.
	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)   $a_n=e^{in\pi /7}$,
	Does the sequence given by ${\displaystyle \frac{n}{2n+1}}$ converge? If so, what is the limit?
	Does the sequence given by $\sqrt{n}$ converge? If so, what is the limit?
	Does the sequence given by ${\displaystyle \frac{1}{\sqrt{n}}}$ converge? If so, what is the limit?
	Does the sequence given by $\sqrt{n+1}-\sqrt{n}$ converge? If so, what is the limit?
	Does the sequence given by $\sqrt{n}(\sqrt{n+1}-\sqrt{n})$ converge? If so, what is the limit?
	Does the sequence given by ${\displaystyle \frac{n}{n^2+1}}$ converge? If so, what is the limit?
	Does the sequence given by ${\displaystyle \frac{2n}{n+1}}$ converge? If so, what is the limit?
	Does the sequence given by ${\displaystyle \frac{3n+1}{2n+5}}$ converge? If so, what is the limit?
	Does the sequence given by ${\displaystyle \frac{n^2-1}{2n^2+3}}$ converge? If so, what is the limit?
	Show that the sequence $a_n={\displaystyle {(1+\frac{1}{n})}^n}$ is increasing and bounded above by 3.
	Let $a\in ℝ$ with $\left|a\right|<1$. Does the sequence given by $a^n$ converge? If so, what is the limit?
	Let $a\in ℝ$ with $a>0$. Does the sequence given by $a^{1/n}$ converge? If so, what is the limit?
	Does the sequence given by $n^{1/n}$ converge? If so, what is the limit?
	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 $\sqrt{a}$.
	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.
	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.
	Let $x_1=1$ and $x_{n+1}=\frac{1}{2+x_n}$. Show that the sequence $x_n$ converges and find the limit.
	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.
	Find the upper and lower limits of the sequence ${\left(-1\right)}^n(1+\frac{1}{n})$.
	Find the upper and lower limits of the sequence given by $a_1=0$, $a_{2k}=\frac{1}{2}{a}_{2k+1}$, and $a_{2k+1}=\frac{1}{2}+a_{2k}$.
	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.
	Let $a_n$ be a bounded sequence. Show that $\lim \inf a_n\le \lim \sup a_n$.
	Let $a_n$ be a bounded sequence. Show that $a_n$ converges if and only if $\lim \sup a_n\le \lim \inf a_n$.
	Let $a_n$ be a bounded sequence such that $\lim \sup a_n\le \lim \inf a_n$. Show that $\lim \sup a_n=\lim \inf a_n=\lim a_n$.
	Let $a_n$ be a real sequence. Show that ${\lim }_{n\to \infty }{a}_n=a$ if and only if $\lim \sup a_n=\lim \inf a_n=a$.
	Prove that ${\displaystyle \sum _{k=1}^n\frac{1}{k(k+1)}=\frac{n}{n+1}}$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)