From 87653cf919fe4541d17be1980e78e0000c774dce Mon Sep 17 00:00:00 2001 From: atusa17 Date: Sat, 29 Sep 2018 13:27:07 -0600 Subject: [PATCH] txs auto checkin --- Documents/LaTeX/Homework 5.tex | 102 +++++++++++++++++++++++++++++++++ 1 file changed, 102 insertions(+) diff --git a/Documents/LaTeX/Homework 5.tex b/Documents/LaTeX/Homework 5.tex index 7c1787d..fac5613 100644 --- a/Documents/LaTeX/Homework 5.tex +++ b/Documents/LaTeX/Homework 5.tex @@ -23,6 +23,7 @@ \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{theorem}{Theorem}[section] +\newtheorem*{theorem*}{Theorem} \newtheorem{corollary}{Corollary}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem*{remark}{Remark} @@ -33,6 +34,107 @@ \begin{document} \maketitle \begin{enumerate} + \item For the following sequences, i) write out the first 5 terms, ii) Use the Monotone Sequence Property to show that the sequences converges. + \begin{enumerate} + \item \textbf{Section 3.3} + \begin{enumerate} + \item[2)] Let $x_1 > 1$ and $x_{n+1} := 2-1/x_n$ for $n \in \N$. Show that $(x_n)$ is bounded and monotone. Find the limit. + \\\\ The first five terms of this sequence are $x_1 >1,x_2 > \frac{3}{2}, x_3 >\frac{4}{5}, x_4 > \frac{5}{4}, x_5 > \frac{6}{5}$. This sequence appears to be decreasing. + \\\\Recall the Monotone Sequence Property: + \begin{theorem*}{Monotone Sequence Property} + A monotone sequence of real numbers is convergent if and only if it is bounded. Further, + \begin{enumerate} + \item If $X=(x_n)$ is a bounded increasing sequence, then + \[\lim (x_n) = \sup \{x_n:n \in \N\}\] + + \item If $Y=(y_n)$ is a bounded decreasing sequence, then + \[\lim (y_n) = \inf \{y_n : n \in \N \}\] + \end{enumerate} + \end{theorem*} + + To show that this sequence converges, we must first find the possible limit points (fixed points) of this sequence. So, + \begin{align*} + x&=2-\frac{1}{x} \\ + x^2 &= 2x -1 \\ + x^2 - 2x + 1 &= 0 \\ + (x-1)^2 &= 0 + \end{align*} + Thus, $x=1$ is a possible limit of this sequence. + \\\\Now, we will prove that $(x_n)$ is bounded by $1$, and since we hypothesized that $(x_n)$ is decreasing, we say that $(x_n)$ is bounded below by 1. + \begin{proof} + We want to show that the sequence $(x_n)$ is bounded below by 1; that is, we want to show that $1 \leq x_n,\ \forall\ n \in \N$. We prove it by method of mathematical induction. + \\\\\textbf{Basis Step:} Let $n=1$. Then + \begin{align*} + x_n &\geq x_{n+1}, &\text{by the definition of decreasing,} \\ + x_1 &\geq x_{1+1} \\ + x_1 &\geq x_2 + \end{align*} + \end{proof} + + \item[3)] Let $x_1 > 1$ and $x_{n+1} := 1 + \sqrt{x_n - 1}$ for $n \in \N$. Show that $(x_n)$ is decreasing and bounded below by $2$. Find the limit. + + \item[7)] Let $x_1 := a>0$ and $x_{n+1} := x_n+1/x_n$ for $n \in \N$. Determine whether $(x_n)$ converges or diverges. + + \item[8)] Let $(a_n)$ be an increasing sequence, $(b_n)$ be a decreasing sequence, and assume that $a_n \leq b_n$ for all $n \in \N$. Show that $\lim (a_n) \leq \lim (b_n)$, and thereby deduce the Nested Intervals Property 2.5.2 from the Monotone Convergence Theorem 3.3.2. + \end{enumerate} + \item $a_1 = 1,\ a_{n+1}=\frac{a_n^2+5}{2a_n}$ + + \item $a_1 = 5,\ a_{n+1}=\sqrt{4+a_n}$ + \end{enumerate} + \item + \begin{enumerate} + \item Show $a_n=\frac{3 \cdot 5 \cdot 7 \cdot \dots (2n-1)}{2 \cdot 4 \cdot 6 \dots (2n)}$ converges to $A$ where $0 \leq A < 1/2$. + + \item Show $b_n = \frac{2 \cdot 4 \cdot 6 \dots (2n)}{3 \cdot 5 \cdot 7 \dots (2n+1)}$ converges to $B$ where $0 \leq B < 2/3$. + \end{enumerate} + + \item \textbf{Section 3.4} + \begin{enumerate} + \item[1)] Give an example of an unbounded sequence that has a convergent subsequence. + + \item[3)] Let $(f_n)$ be the Fibonacci sequence of Example 3.1.2(d), and let $x_n := f_{n+1}/f_n$. Given that $\lim (x_n) =L$ exists, determine the value of $L$. + + \item[4a)] Show that the sequence $(1-(-1)^n+1/n)$ converges. + + \item[16)] Give an example to show that Theorem 3.4.9 fails if the hypothesis that $X$ is a bounded sequences is dropped. + + \item[18)] Show that if $(x_n)$ is a bounded sequence, then $(x_n)$ converges if and only if $\lim \sup (x_n) = \lim \inf (x_n)$. + + \item[19)] Show that if $(x_n)$ and $(y_n)$ are bounded sequences, then + \[\lim \sup (x_n + y_n) \leq \lim \sup (x_n) + \lim \sup (y_n).\] + Give an example in which the two sides are not equal. + \end{enumerate} + \item + \begin{enumerate} + \item Show that $x_n=e^{\sin (5n)}$ has a convergent subsequence. + + \item Give an example of a bounded sequence with three subsequences converging to three different numbers. + + \item Give an example of a sequence $x_n$ with $\lim \sup x_n = 5$ and $\lim \sup x_n = -3$. + + \item Let $\lim \sup x_n = 2$. True or False: if $n$ is sufficiently large, then $x_n > 1.99$. + + \item Compute the infimum, supremum, limit infimum, and limit supremum for $a_n = 3 - (-1)^n - (-1)^n/n$. + \end{enumerate} + + \item + \begin{enumerate} + \item If $a_n$ and $b_n$ are strictly increasing, then $a_n + b_n$ is strictly increasing. + + \item If $a_n$ and $b_n$ are strictly increasing, then $a_n \cdot b_n$ is strictly increasing. + + \item If $a_n$ and $b_n$ are monotonic, then $a_n + b_n$ is monotonic. + + \item If $a_n$ and $b_n$ are monotonic, then $a_n \cdot b_n$ is monotonic. + + \item If a monotone sequence is bounded, then it is convergent. + + \item If a bounded sequence is monotone, then it is convergent. + + \item If a convergent sequence is monotone, then it is bounded. + + \item If a convergent sequence is bounded, then it is monotone. + \end{enumerate} \end{enumerate} \end{document} \ No newline at end of file