Problem Set - Sequences

## Problem Set - 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 ${\mathrm{lim}}_{n\to \infty }{a}_{n}$ is unique. Prove that if $\left({a}_{n}\right)$ converges then $\left({a}_{n}\right)$ is bounded. Prove that if ${\mathrm{lim}}_{n\to \infty }{a}_{n}=a$ and ${\mathrm{lim}}_{n\to \infty }{b}_{n}=b$ then ${\mathrm{lim}}_{n\to \infty }{a}_{n}+{b}_{n}=a+b$. Prove that if ${\mathrm{lim}}_{n\to \infty }{a}_{n}=a$ and ${\mathrm{lim}}_{n\to \infty }{b}_{n}=b$ then ${\mathrm{lim}}_{n\to \infty }{a}_{n}{b}_{n}=ab$. Prove that if ${\mathrm{lim}}_{n\to \infty }{a}_{n}=a$ and ${\mathrm{lim}}_{n\to \infty }{b}_{n}=b$ and ${b}_{n}\ne 0$ for all $n\in {ℤ}_{>0}$ then ${\mathrm{lim}}_{n\to \infty }\frac{{a}_{n}}{{b}_{n}}=\frac{a}{b}$. Prove that if ${\mathrm{lim}}_{n\to \infty }{a}_{n}=\ell$ and ${\mathrm{lim}}_{n\to \infty }{c}_{n}=\ell$ and ${a}_{n}\le {b}_{n}\le {c}_{n}$ for all $n\in {ℤ}_{>0}$ then ${\mathrm{lim}}_{n\to \infty }{b}_{n}=\ell$. 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)   ${a}_{n}=\frac{1}{n}$, (c)   ${a}_{n}=\frac{{\left(n\text{!}\right)}^{2}{5}^{n}}{\left(2n\right)\text{!}}$, (d)   ${a}_{1}=3$, ${a}_{n}=\frac{1}{2}\left({a}_{n-1}+\frac{5}{{a}_{n-1}}\right)$, (e)   ${a}_{n}={\left(1+\frac{1}{n}\right)}^{n}$, (f)   ${a}_{n}={e}^{in\pi /7}$, Does the sequence given by $\frac{n}{2n+1}$ 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 $\frac{1}{\sqrt{n}}$ 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 $\sqrt{n}\left(\sqrt{n+1}-\sqrt{n}\right)$ converge? If so, what is the limit? Does the sequence given by $\frac{n}{{n}^{2}+1}$ converge? If so, what is the limit? Does the sequence given by $\frac{2n}{n+1}$ converge? If so, what is the limit? Does the sequence given by $\frac{3n+1}{2n+5}$ converge? If so, what is the limit? Does the sequence given by $\frac{{n}^{2}-1}{2{n}^{2}+3}$ converge? If so, what is the limit? Show that the sequence ${a}_{n}={\left(1+\frac{1}{n}\right)}^{n}$ is increasing and bounded above by 3. Let $a\in ℝ$ with $|a|<1$. 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 ${x}_{n+1}=\frac{1}{2}\left({x}_{n}+a/{x}_{n}\right)$. 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 ${x}_{n+1}=\frac{1}{7}\left({x}_{n}^{3}+2\right)$. 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 ${\left(-1\right)}^{n}\left(1+\frac{1}{n}\right)$. 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 $\mathrm{inf}{a}_{n}$, $\mathrm{lim}\mathrm{inf}{a}_{n}$, $\mathrm{lim}\mathrm{sup}{a}_{n}$, and $\mathrm{sup}{a}_{n}$ are equal. Let ${a}_{n}$ be a bounded sequence. Show that $\mathrm{lim}\mathrm{inf}{a}_{n}\le \mathrm{lim}\mathrm{sup}{a}_{n}$. Let ${a}_{n}$ be a bounded sequence. Show that ${a}_{n}$ converges if and only if $\mathrm{lim}\mathrm{sup}{a}_{n}\le \mathrm{lim}\mathrm{inf}{a}_{n}$. Let ${a}_{n}$ be a bounded sequence such that $\mathrm{lim}\mathrm{sup}{a}_{n}\le \mathrm{lim}\mathrm{inf}{a}_{n}$. Show that $\mathrm{lim}\mathrm{sup}{a}_{n}=\mathrm{lim}\mathrm{inf}{a}_{n}=\mathrm{lim}{a}_{n}$. Let ${a}_{n}$ be a real sequence. Show that ${\mathrm{lim}}_{n\to \infty }{a}_{n}=a$ if and only if $\mathrm{lim}\mathrm{sup}{a}_{n}=\mathrm{lim}\mathrm{inf}{a}_{n}=a$. Prove that $\sum _{k=1}^{n}\frac{1}{k\left(k+1\right)}=\frac{n}{n+1}$.