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\begin{document}

\title{Homework \#10, Real Analysis I, Fall 2012}


\author{Marius Ionescu}
\date{\bf{Due Sunday, 12/09/2012}; You do not have to type this
  assignment using Latex}
\maketitle
\begin{problem}[Exercise 6.2.3 on page 161]
Consider the sequence of functions
\[
h_n(x) = \frac{x}{1+x^n}
\]
over the domain $[0,\infty)$.
\begin{itemize}
\item[(a)] Find the pointwise limit of $(h_n)$ on 
  $[0,\infty)$.
  \item[(b)] Explain how we know that the convergence 
    \textit{cannot} be uniform on $[0,\infty)$.
    \item[(c)] Choose a smaller set over which the convergence is 
uniform and supply an argument to show that this is indeed the 
case.
\end{itemize}
\end{problem}

\begin{problem}[Exercise 6.2.7 on page 161]
Assume that $(f_n)$ converges uniformly to $f$ on $A$ and that each
$f_n$ is uniformly continuous. Prove that $f$ is uniformly continuous
on $A$.
\end{problem}
%\begin{proof}
  
%\end{proof}
\begin{problem}[Exercise 6.3.1 on page 166]
  \begin{itemize}
  \item[(a)] Let
    \[
    h_n(x)=\frac{\sin(nx)}{n}.
    \]
    Show that $h_n\to 0$ uniformly on $\R$. At what points does the
    sequence of derivatives $h_n^\prime$ converge?
    \item[(b)] Modify this example to show that it is possible for a
      sequence $(f_n)$ to converge uniformly but for $(f_n^\prime)$ to
      be unbounded.
  \end{itemize}
\end{problem}
\begin{problem}[Exercise 6.4.3 on page 168]
  \begin{itemize}
  \item[(a)] Show that $g(x)=\sum_{n=1}^\infty \cos(2^nx)/2^n$ is
    continuous on all of $\R$.
  \item[(b)] Prove that $h(x)=\sum_{n=1}^\infty x^n/n^2$ is continuous
    on $[-1,1]$.
  \end{itemize}
\end{problem}
\begin{problem}[Exercise 6.5.1 on page 174]
  Consider the function $g$ defined by the power series
  \[
  g(x)=x-\frac{x^2}2+\frac{x^3}3-\frac{x^4}4+\frac{x^5}5-\cdots.
  \]
  \begin{itemize}
  \item[(a)] Is $g$ define on $(-1,1)$? Is it continuous on this
    set? Is $g$ defined on $(-1,1]$? Is it continuous on this set? What
    happens on $[-1,1]$? Can the power series for $g(x)$ possible
    converge for any other points $\vert x\vert >1$? Explain.
  \item[(b)] For what values of $x$ is $g^\prime(x)$ defined? Find a
    formula for $g^\prime$.
  \end{itemize}
\end{problem}
\begin{problem}[Exercise 6.5.2 on page 174]
  Find suitable coefficients $(a_n)$ so that the 
resulting power series $\sum a_n x^n$
\begin{itemize}
\item[(a)] converges absolutely for all $x\in[-1,1]$ and 
diverges off of this set;
\item[(b)] converges conditionally at $x=-1$ and diverges 
at $x=1$;
\item[(c)] converges conditionally at both $x=-1$ and 
$x=1$.
\item[(d)] Is it possible to find an example of a power series 
that converges conditionally at $x=-1$ and converges absolutely
at $x=1$?
\end{itemize}
\end{problem}
\end{document}
