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\begin{document}

\title{Homework \#5, Real Analysis I, Fall 2012}


\author{Marius Ionescu}
\date{\bf{Due Friday, 10/19/2012}}
\maketitle
\begin{problem}[Exercise 2.7.1 on page 67; \textbf{This problem is
    worth double the points (12points)}]
Proving the Alternating Series Test (Theorem 2.7.7) amounts to showing
that the sequence of partial sums
\[
s_n=a_1-a_2+a_3-\dots\pm a_n
\]
converges. Different characterization of completeness lead to
different proofs. {\bf Please choose one of the following methods to
  prove the Alternating Series Test}:
\begin{enumerate}
\item[(a)] Prove the Alternating Series Test by showing that $(s_n)$
  is a Cauchy sequence.
\item[(b)] Prove the Alternating Series Test using the Nested Interval
  Property (Theorem 1.4.1)
\item[(c)] Consider the subsequences $(s_{2n})$ and $(s_{2n+1})$, and
  show how the Monotone Convergence Theorem leads to a third proof for
  the Alternating Series Test.
\end{enumerate}
\end{problem}

\begin{problem}[Exercise 2.7.4 on page 68]
  Give an example to show that it is possible for both
$\sum x_n$ and $\sum y_n$ to diverge but for $\sum x_ny_n$ to 
converge.
\end{problem}
%\begin{proof}
  
%\end{proof}
\begin{problem}[Exercise 2.7.5 on page 68]
\begin{itemize}
\item[(a)]  Show that if $\sum a_n$ converges absolutely, then 
$\sum a_n^2$ also converges absolutely. Does this proposition hold without 
absolute convergence?
\item[(b)] If $\sum a_n$ converges and $a_n\geq 0$, can we conclude 
  anything about $\sqrt{a_n}$?
\end{itemize}
\end{problem}
\begin{problem}[Exercise 2.7.9 on page 68 (\textbf{Ratio Test})] Given a series 
$\sum_{n=1}^{\infty} a_n$ with $a_n \neq 0$, the Ratio Test states that
if $(a_n)$ satisfies 
$$
\lim\left|\frac{a_{n+1}}{a_n}\right| = r < 1\ ,
$$
then the series converges absolutely.
\begin{itemize}
\item[(a)] Let $r'$ satisfy $r<r'<1$. (Why must such an $r'$ exist?)
Explain why there exists an $N$ such that $n\geq N$ implies 
$|a_{n+1}|\leq |a_n|r'$.
\item[(b)] Why does $|a_N|\sum(r')^n$ necessarily converge?
\item[(c)] Now, show that $\sum |a_n|$ converges.
\end{itemize}
\end{problem}
\begin{problem}[This problem is not from the textbook]
Test each of the following series for absolute convergence,
conditional convergence, or divergence:
\begin{itemize}
\item[(a)] \[ \sum_{n=1}^\infty \frac{(-1)^{n+1}n}{n^2-5n+1} \]
\item[(b)] \[ \sum_{n=1}^\infty \frac{(-1)^{n+1}(n-5)}{n^3-7n-9}\]
\item[(c)] \[ \sum_{n=1}^\infty \frac{(-1)^{n+1}n}{n+1}\]
\end{itemize}
\end{problem}
\end{document}
