% HMC Math dept HW template example
% v0.04 by Eric J. Malm, 10 Mar 2005
\documentclass[12pt,letterpaper,boxed]{hmcpset}

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\usepackage[margin=1in]{geometry}

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\usepackage{graphicx}

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\renewcommand{\labelenumi}{(\alph{enumi})}

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\name{}
\class{Math 315}
\assignment{Homework \#17}
\duedate{6/9/2017}

% macros
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\II}{\mathbb{I}}

\begin{document}

\problemlist{Abbott: 6.3.1, 6.4.3}


\begin{problem}[6.3.1]
Consider the sequence of functions defined by
\[ g_n(x) = \frac{x^n}{n}. \]
\begin{enumerate}
\item Show $(g_n)$ converges uniformly on $[0,1]$ and find $g = \lim g_n$. Show that $g$ is differentiable and compute $g'(x)$ for all $x \in [0,1]$.
\item Now, show that $(g_n')$ converges on $[0,1]$. Is the convergence uniform? Set \linebreak $h = \lim g_n'$ and compare $h$ and $g'$. Are they the same?
\end{enumerate}
\end{problem}

%\begin{solution}
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%\end{solution}

\newpage

\begin{problem}[6.4.3]
\begin{enumerate}
\item Show that
\[ g(x) = \sum_{n=0}^{\infty} \frac{\cos(2^nx)}{2^n} \]
is continuous on all of $\RR$.
\item The function $g$ was cited in Section 5.4 as an example of a continuous nowhere differentiable function. What happens if we try to use Theorem 6.4.3 to explore whether $g$ is differentiable?
\end{enumerate}
\end{problem}

%\begin{solution}
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%\end{solution}


\end{document}
