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\title{\vspace*{-1.5in}\textbf{\S8.4: Part I - Alternating 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{March 5, 2018}
\chead{\S8.4: Alt. Series}
\vspace{-.8in}
\noindent
\textbf{Key Points:}
\begin{itemize}
\item If the terms in a series alternate signs, we call the series an \textbf{alternating series}. 
\item An alternating series can be written in the form 
\[
\sum_{n=1}^\infty (-1)^{n-1}b_n \quad \text{or}\quad \sum_{n=1}^\infty (-1)^{n}b_n, \quad \text{where $b_n \geq 0$}.
\]
(i.e. $b_n$ includes no negative terms)
\item \textbf{Alternating series test}:\begin{flushleft}\framebox{\begin{minipage}[t]{.6\textwidth}\vspace*{1.3in}\hspace*{3cm} \end{minipage}}\end{flushleft}

\item Note: Recall that to show $b_n$ is decreasing, show
\[
\rule{1in}{1pt} \quad \text{or} \quad \rule{1in}{1pt} \quad \text{or} \quad \rule{1in}{1pt} 
\]
\item Note: If in an alternating series, $\lim_{n\to\infty} b_n \neq 0$, then\\[24pt]
\item Alternating series remainder test: If $\sum(-1)^nb_n$ converges by the alternating series test, then 
\[
|\text{Error}| = |R_n| = \phantom{laskdjflasdkjfasdfasd}
\]
\end{itemize}
\textbf{Examples:}
\begin{enumerate}
\item (Alternating Harmonic Series) Does $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}$ converge or diverge?\newpage
\item Does $\displaystyle \sum_{n=2}^\infty \frac{(-1)^{n}\ln{n}}{n}$ converge or diverge?\vfill
\item Does $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n}n!}{e^n}$ converge or diverge?\vfill
\item Estimate $\displaystyle \sum_{n=1}^\infty  (-1)^{n+1} \frac{1}{n^5}$ using three terms. How accurate is your estimate?\vfill
\item How many terms should we add to ensure that our estimate of $\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}$ is within $0.0001$ of the true value? \vfill
\end{enumerate}

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