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% THEOREM ENVIRONMENTS
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\newtheorem*{theorem}{{Theorem}}
\newtheorem*{lemma}{{Lemma}}
\newtheorem*{proposition}{{Proposition}}
\newtheorem*{definition}{{Definition}}
\newtheorem*{corollary}{{Corollary}}
\newtheorem*{example}{{Example}}
\newtheorem*{remark}{{Remark}}
\newtheorem*{claim}{{Claim}}
\newtheorem*{question}{{Question}}

\theoremstyle{plain}
\newtheorem*{reminder}{{\textit{Reminder}}}
\newtheorem*{hint}{{\textit{Hint}}}

% ENVIRONMENTS FOR CITING BOOK STATEMENTS
\newcommand{\prop}[1]{\paragraph{Proposition #1}}
\newcommand{\axm}[1]{\paragraph{Axiom #1}}
\newcommand{\thm}[1]{\paragraph{Theorem #1}}


% COMMANDS
\newcommand{\solution}{{\color{blue} \textsc{\ \\ Solution.\ \ }}}
\newcommand{\revisedsolution}{{\color{blue} \textsc{\ \\ Revised Solution.\ \ }}}
\newcommand{\Z}{\mathbb{Z}} % the integers
\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\R}{\mathbb{R}} % real nubers
\newcommand{\Q}{\mathbb{Q}} % rational numbers
\newcommand{\divides}{\ \vert \ }
\newcommand{\st}{ \ : \ } % such that
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\newcommand{\img}{\mathrm{im}} % image
\newcommand{\into}{\hookrightarrow} % injection
\newcommand{\onto}{\twoheadrightarrow} % surjection
\newcommand{\map}[1]{\xrightarrow{#1}} % named map
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% SOME ENVIRONMENTS

\newcommand{\exheading}[1]{\section*{Exercise #1}}
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% \setcounter{exnum}{1} % default 0 start
\newcommand{\exercise}{
    \stepcounter{exnum} 
    \exheading{\theexnum}
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\newcommand{\bea}{\begin{enumerate}[label={(\alph*)}]}
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%%%%
\title{MAT 108: Problem Set 8}
\author{(ADD NAME)}
\date{Due 3/7/23 at 11:59 pm on Canvas}

\begin{document}

\maketitle


\paragraph{Reminders:}
\begin{itemize}

%\item Put your name at the top!
\item Your homework submission must be typed up in full sentences, with proper mathematical formatting. Handwritten homework submissions will receive a score of 0. Solutions containing incomplete sentences or poor formatting will lose points.
\item You will receive feedback on PS6 by next Tuesday, 3/7. PS6 revisions are due Friday, 3/10 at 11:59 pm. Underneath your old solution, type 
\vspace{-3mm}
    \begin{verbatim}
        \revisedsolution
    \end{verbatim}
\vspace{-7mm}
and then type your revised solution.
\end{itemize}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise 

Determine the sum of the series \[\sum_{n=0}^\infty \frac{2^n}{3^{n+1}}.\]

\noindent \emph{Be careful. You may use the formula for the sum of a geometric series, but be careful with indices, and make sure you know what the ``first term'' is.}


\solution


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise 


Prove Proposition 12.4:

\begin{proposition}
    Let $\displaystyle (d_k)_{k=1}^\infty$ be a sequence of digits. Then $\displaystyle \sum_{j=1}^\infty d_j \cdot 10^{-j}$ converges. 
\end{proposition}

\begin{remark}
The digits in this exercise are base 10 digits, i.e.\ for all $j$, $d_j \in \{0, 1, \ldots, 9\}$. 
\end{remark}



\solution



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise 

Write the following decimal expansions as fractions.
\bea
\item $5.\overline{6} = 5.66666...$
\item $0.346\overline{127} = 0.346127127127...$
\ee

\solution


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise 

Express the following fractions as decimals.
\bea
\item $\frac{71}{13}$
\item $\frac{34}{31}$
\ee

\solution







\end{document}
