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\title{\vspace*{-1.5in}\textbf{\S8.2: Series}\\\normalsize (Thanks to Faan Tone Liu)}
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\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\rhead{Feb. 23, 2018}
\chead{\S8.2: Series}
\vspace{-.8in}
\noindent
\textbf{Key Points (Part I):}
\begin{itemize}
\item An \textbf{infinite series} is the sum of the terms of a sequence:\vskip2cm
\item More precisely, an infinite series is related to a special sequence of \textbf{partial sums}:\vskip4cm
\item This allows us to see that the sum of an infinite series is:\vskip5cm
\item Graphical perspective (infinite series are related to Riemann sums):\vskip4cm
\item Important tool:\begin{center}\framebox{\underline{\textbf{Divergence Test:}}\begin{minipage}[t]{.6\textwidth}\vspace*{1in}\hspace*{3cm} \end{minipage}}\end{center}
\end{itemize}

\newpage
\noindent
\textbf{Examples:}\vskip12pt
\begin{enumerate}
\item (Using the divergence test) Does $\displaystyle \sum_{n=1}^\infty \frac{e^n}{n^2}$ converge or diverge?\vskip.8in
\item (Harmonic series) Does $\displaystyle \sum_{n=1}^\infty\frac{1}{n}$ converge or diverge?\vskip2.5in
\item (Telescoping Series) Explicitly calculate the sum of the series $\displaystyle \sum_{i=1}^\infty\frac{1}{i(i+1)}$. 
\end{enumerate}\newpage
\noindent
\textbf{Key Points (PartII):}
\begin{itemize}
\item $\displaystyle \sum_{n=1}^\infty a_n = \underline{\phantom{blahahahahahahahah}}$. 
\item The goal is to determine if $\displaystyle \sum_{n=1}^\infty a_n$ converges or diverges. So far, we have a few tools:
\begin{itemize}
\item \textbf{Divergence test}. Check if $\underline{\phantom{asldkfjalsdkfjalsdkfj}}$. If the limit is not zero, you are done and $\displaystyle \sum_{n=1}^\infty a_n$ \underline{\phantom{alsdkjflaskdjfldskjflksjdf}}. If $a_n \to 0$, then too bad, we have to do more.\\
\item We can directly calculate the partial sums $\displaystyle S_N = \sum_{n=1}^N a_n$ for \textbf{telescoping series} and take the limit $\displaystyle \lim_{N\to \infty} S_N$ to establish convergence or divergence.\\
\item \textbf{Geometric series} are our friends! A geometric series has the form \underline{\phantom{alsdkjfladjdf}}. We know that
\begin{itemize}
\item $\displaystyle \sum_{i=1}^n ar^{i-1} = \underline{\phantom{alsdkjflaskdjfldsk}}$. In other words, \\[10mm]
\item If $|r| < 1$, then $\displaystyle \sum_{i=1}^\infty ar^{i-1} = \underline{\phantom{alsdkjflassdfdfd}}$. In other words,\\[10mm]
\item If $|r| \geq 1$, then $\displaystyle \sum_{i=1}^\infty ar^{i-1}$ \underline{\phantom{alsdkjflassdfdfd}}.\\
\end{itemize}
\end{itemize}
\item The \textbf{harmonic series} is \underline{\phantom{alsdkjflaskdjfldskjflksjdf}}. It \underline{\phantom{alsdkjflaskdjfldskjflksjdf}}.
\item Other notes:
\end{itemize}
\newpage
\noindent
\textbf{Examples:} For each of these series, write it in expanded form if it is given in $\Sigma$-notation, and in $\Sigma$-notation if it is given in expanded form. Then, determine if the series converges and if so, find the sum. \vskip12pt
\begin{tabular}{l@{\hspace{.5in}}|l}
\textbf{Ex A.}\quad $\displaystyle \sum_{n=0}^\infty\left(\frac{\pi}{e}\right)^{n-1}$ & \textbf{Ex B.}\quad $\displaystyle \sum_{i=1}^\infty \ln\left(\frac{i+1}{i}\right)$\\[2.2in]\hline&\\
\textbf{Ex C.}\quad $\displaystyle 3 + \frac{3}{2} + 1 + \frac{3}{4} + \frac{3}{5} + \frac{3}{6} + \cdots$ & \textbf{Ex D.}\quad $\displaystyle \sum_{n=2}^\infty 3\left(\frac{2}{3}\right)^{n-1}$\\[2.2in]\hline&\\
\textbf{Ex E.}\quad $\displaystyle 5 + \sqrt{5} + \sqrt[3]{5} + \sqrt[4]{5} + \cdots$ & \textbf{Ex F.}\quad $\displaystyle 1 + x + x^2 + x^3 + \cdots$\\[2in]
\end{tabular}
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