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\title{Math 3110: Homework 5}
%\author{Zach Norwood}
\date{Due Wednesday, 27 February 2019}

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\begin{document}
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\noindent
You will be graded on both the accuracy and the clarity of your solutions.
One purpose of the homework is to give you an opportunity to practice your proof-writing skills.

You are welcome\,---\,encouraged, even!\,---\,to collaborate on homework, but you should not 
copy solutions from any source, nor should you submit anything that you don't understand.

\begin{problem}
 Let $r\in(-1,1)$.
 Show directly (without appealing to recent theorems from lecture)
 that the sequence of partial sums of the geometric series $\sum_{n\ge0} r^n$ is Cauchy.
\end{problem}

\begin{problem} \hfill
 \begin{enumerate} [(a)]
 \item [($\checkmark$)] Show (but \emph{do not} turn in with your homework)
 that changing finitely many terms of a series does not affect its convergence.
 That is, if $\sum a_n$ converges and there is $N\in\N$ such that $n\ge N$ implies $b_n = a_n$,
 then $\sum b_n$ converges too.
 \item Prove that if $(na_n)_{n\in\N}$ converges to a nonzero real number $L$,
 then the series $\sum a_n$ diverges. (\emph{Hint}: Comparison.)
 Give an example to show that the converse is false.
 
 \item Prove that if $(n^2 a_n)_{n\in\N}$ converges (to any real number),
 then the series $\sum a_n$ converges. (\emph{Hint}: Comparison.)
 Give an example to show that the converse is false.
 \end{enumerate}
\end{problem}

\begin{problem}
 Show how to construct a rearrangement of the alternating harmonic series that
 converges to $7$.
\end{problem}

\begin{problem}
 The \textbf{Alternating Series Test} is the following
 fact: if $(a_n)_{n\in\N}$ is a sequence of nonnegative terms converging to $0$, then the alternating
 series $\sum_{n\ge0} (-1)^n a_n$ converges.
 By imitating our proof of the convergence of $\sum_{n\ge1} \tfrac{(-1)^{n+1}}{n}$ or otherwise,
 prove the Alternating Series Test.
\end{problem}

\begin{problem}
 Prove that if $\sum_{n\ge0} a_n$ converges absolutely to $L$,
 then any rearrangement of $\sum_{n\ge0} a_n$ also converges to $L$.
 
 Indicate clearly in your proof where you use the fact that the series is absolutely convergent.
 
 (\emph{Hint}: Consider the partial sums $(s_n)$ of $\sum a_n$ and the partial sums $(s_n')$ of the 
 rearrangement, and show that $\displaystyle\lim_{n\to\infty} \abs{s_n-s_n'}=0$.)
 \end{problem}
 
\begin{problem}
 Prove the ``$p$-test'': the series $\sum_{n\ge1} \tfrac{1}{n^p}$ converges if $p > 1$ and diverges
 if $p \le 1$.
 
 (\emph{Hint}: Use Homework 4, Problem 4.)
\end{problem}

\begin{problem}
 Prove the Ratio Test: Suppose $(a_n)_{n\in\N}$ is a sequence converging to $0$ and set
 $\rho = \limsup \abs{\tfrac{a_{n+1}}{a_n}}$. If $\rho < 1$, then $\sum a_n$ is absolutely convergent.
 
 (For the definition of limsup, see Homework 4, Problem 5.)
\end{problem}





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