\documentclass{amsart}[20pt]
\usepackage[top=2cm,bottom=2cm]{geometry}
\usepackage{amssymb,mathrsfs}
\geometry{top=1.5cm}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\eps}{\varepsilon}
\pagenumbering{gobble}
\theoremstyle{definition}
\newtheorem{que}{Question}[section]
\renewcommand{\thesection}{\Alph{section}}
\title{MTH435 - Homework 6}

\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 11/1/17.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Warm-up Questions}
 
 \begin{que}
  Compare the converge for $\sum_{n=1}^\infty a_n$ and $\sum_{k=0}^\infty 2^k a_{2^k}$ for the following.
  \begin{itemize}
   \item[(a)] $a_n = \frac{1}{n}$.
   \item[(b)] $a_n = \frac{1}{n^2}$. 
  \end{itemize}
\end{que}
  
  
 \begin{que} Prove the following assertions from class.
 \begin{itemize}
  \item[(a)] If $\limsup |x_n|^{\frac{1}{n}} = r$ and $r<1$ then $\sum x_n$ converges absolutely.
  \item[(b)] If $\limsup \left| \frac{x_{n+1}}{x_n} \right| = r$ and $r < 1$ then $\sum x_n$ converges absolutely.
 \end{itemize}
 
  
 \end{que}

\begin{que}
 Discuss the convergence or divergence of $\sum x_n$ for the following series.
 \begin{itemize}
  \item[(a)] $x_n = \frac{\sqrt{n+1} - \sqrt{n}}{n}$.
  \item[(b)] $x_n = \frac{1}{n \log n}$.
  \item[(c)] $x_n = \frac{n}{2^n}$.
  \item[(d)] $x_n = \frac{n^n}{e^n}$.
  \item[(e)] $x_{2n-1} = \frac{1}{(2n-1)^2}$ and $x_{2n} = \frac{1}{(2n)^3}$.
 \end{itemize}

\end{que}
 


 
 
 
 
 
 
 
 
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Submitted Questions}
 
 \begin{que}
  Suppose $(a_n)$ is a decreasing sequence of positive numbers. Define
  \[
   s_n = \sum_{i=1}^n a_i \quad \text{ and } \quad t_k = \sum_{j=0}^k 2^j a_{2^j}.
  \]
  \begin{itemize}
   \item[(a)] Prove that if $n < 2^k$ then $s_n \leq t_k$.
   \item[(b)] Prove that if $n > 2^k$ then $2s_n \geq t_k$.
   \item[(c)] Deduce that $\sum_{n=1}^\infty a_n$ converges if and only if $\sum_{k=0}^\infty 2^k a_{2^k}$ converges.
  \end{itemize}

 \end{que}

 \begin{que}
  For the following sequences, decide whether $\sum a_n$ converges or diverges.
  \begin{itemize}
   \item[(a)] $a_n = \frac{e^n}{2^{2n}}$.
   \item[(b)] $a_n = \frac{1}{n (\log n)^2}$
   \item[(c)] $a_n = \frac{(n!)^2}{(2n)!}$.
  \end{itemize}

 \end{que}

  \begin{que}
  Define the sequence $(x_n)$ by $x_{2n-1} = \frac{1}{2^n}$ and $x_{2n} = \frac{1}{3^n}$.
  \begin{itemize}
   \item[(a)] Compute $\liminf |x_n|^{\frac{1}{n}}$ and $\limsup |x_n|^{\frac{1}{n}}$.
   \item[(b)] Compute $\liminf \left| \frac{x_{n+1}}{x_n} \right|$ but show that $\left| \frac{x_{n+1}}{x_n} \right|$ is not bounded above.
   \item[(c)] Conclude that the ratio test is inconclusive for the series $\sum x_n$ but the root test does work.
  \end{itemize}



 \end{que}

 
 
 
 
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Challenge Questions}
 
 \begin{que}
  We prove the root test is stronger than the ratio test. That is, if the ratio test shows convergence then the root test also does; on the other hand, if the root test is inconclusive, then the ratio test is also inconclusive. In fact, we will prove that for any sequence of non-zero numbers
  \[
   \limsup |x_n|^{\frac{1}{n}} \leq \limsup \left| \frac{x_{n+1}}{x_n} \right|. 
  \]
\begin{itemize}
 \item[(a)] Show that if $r > \limsup \left| \frac{x_{n+1}}{x_n} \right|$ then there exists $N \in \N$ such that $\left|  \frac{x_{n+1}}{x_n} \right| \leq r$ for all $n \geq N$.
 \item[(b)] Deduce that $x_n \leq r^{n-N}x_N$ for all $n \geq N$.
 \item[(c)] By taking $n$th roots, show that $\limsup |x_n|^{\frac{1}{n}} \leq r$ and so $\limsup |x_n|^{\frac{1}{n}} \leq \limsup \left| \frac{x_{n+1}}{x_n} \right|$.
\end{itemize}

  
 \end{que}

 
 
 
 
 
  \end{document}