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\title{\vspace*{-1.5in}\textbf{Math 2300--007: Exam II Study Guide}}
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\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\chead{March 10, 2018}
\rhead{Exam II Study Guide}
\vspace{-.5in}
\noindent
\textbf{Topics List:}
\begin{itemize}
\item \S6.6: Applications of Integration
\begin{itemize}
\item Work
\begin{itemize}
\item Springs and Hooke's Law
\item Rope/chain lifting
\item Water pumping problems
\end{itemize}

\item Center of Mass
\begin{itemize}
\item Moments $M_x$ and $M_y$
\end{itemize}
\item Tips: Draw a small rectangle ($dx$ or $dy$), draw a slice, and put a coordinate system on the picture.
\end{itemize}

\item \S8.1: Sequences
\begin{itemize}
\item ``Connect the dots'' function
\item Limits involving $\infty$
\item L'H\^{o}pital's Rule
\item Squeeze Theorem
\item Alternating sequences (look at the sequence $|a_n|$)
\item Tip: Identify the ``dominant'' function and factor it out
\end{itemize}

\item \S8.2: Intro to Series
\begin{itemize}
\item Definition of a series
\item Partial sums (the sum of a series is the limit of the partial sums)
\item Divergence Test (if terms don't go to zero, the series diverges)
\item Telescoping series
\item Geometric series
\end{itemize}

\item \S8.3: Comparison Tests (need positive terms!)
\begin{itemize}
\item Integral Test
\item $p$-series (if $p \leq 1$, $\sum\frac{1}{n^p}$ diverges, if $p > 1$, $\sum\frac{1}{n^p}$ converges)
\item Direct Comparison Test
\item Limit Comparison Test
\end{itemize}

\item \S8.4: Other Convergence Tests
\begin{itemize}
\item Alternating series and Alternating Series Test
\item Absolute convergence
\item Ratio Test
\end{itemize}

\end{itemize}
\newpage
\noindent
\textbf{Selected Example Problems Provided by Class for \S6.6,8.1,8.2,8.3:}\vskip12pt
\begin{enumerate}
\item Find The work if a rope weighing $4$lbs/ft hoists a $300$lb weight up the side of a building $500$ feet tall.
\item An inverted conical tank with height 20m and base diameter 25m contains oil with density 800kg/$\text{m}^3$. The height of the oil is 10m. How much work is done pumping all of the oil out of the top? \vspace*{-12pt}
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\draw[|-|] (1,0)--(1,.5) node[right]{$10$m}--(1,1);
\draw[|-|] (-1.2,0)--(-1.2,1) node[left]{$20$m}--(-1.2,2);
\draw[|-|] (1,2.25)--(0,2.25) node[above]{$25$m}--(-1,2.25);
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\item Problems 19 and 21 from page 473 of the course text. These are water pumping problems with pictures I couldn't easily replicate.

\item Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then, find the exact coordinates of the centroid.
\begin{enumerate}[(a)]
\item $y = 4-x^2,\ y=0$
\item $3x+2y = 6,\ y=0,\ x=0$
\end{enumerate}

\item Consider the sequence $-\frac{5}{2}, \frac{8}{4}, \frac{-11}{8}, \frac{14}{16},\ldots, a_n, \ldots$. Find $a_n$. Does this sequence converge or diverge?

\item  Suppose $a_n = \frac{\cos{n}}{n^2}$. Find $\displaystyle \lim_{n\to \infty}a_n$. 

\item Do the following series converge or diverge? How do you know? Can you figure out the exact sum? 
\begin{enumerate}[(a)]
\item $\displaystyle \sum_{n=1}^\infty \frac{n^2 + 3n+1}{2n^2+4}$
\item $\displaystyle\sum_{n=1}^\infty \ln\left(\cos{n}\right)$
\item $\displaystyle\sum_{n=2}^\infty \left(\frac{2}{3}\right)^n$
\item $\displaystyle\sum_{n=0}^\infty 8\left(\frac{1}{3}\right)^n$
\item $\displaystyle\sum_{n=1}^\infty\left(\frac{1}{n} - \frac{1}{n+1}\right)$
\item $\displaystyle\sum_{n=0}^\infty \left(\frac{1}{n+1} - \frac{1}{n+2}\right)$
\item $\displaystyle\sum_{n=1}^\infty \frac{1}{\sqrt[5]{n}}$
\item $\displaystyle\sum_{n=1}^\infty \frac{1}{\sqrt{n+4}}$
\item $\displaystyle\sum_{n=1}^\infty \left(\frac{1}{n^2} + 1\right)^2$
\item $\displaystyle\sum_{n=4}^\infty \frac{n^2}{n^3-3}$
\item $\displaystyle\sum_{n=1}^\infty \frac{\sqrt{2n^2 + 4n + 1}}{n^3+9}$
\item $\displaystyle\sum_{n=1}^\infty \frac{4n^2-n}{n^3+9}$
\item $\displaystyle \sum_{n=0}^\infty \frac{2^n\sin^2(5n)}{4^n+\cos^2(n)}$
\item $\displaystyle \sum_{n=3}^\infty \frac{e^{-n}}{n^2 + 2n}$
\end{enumerate}
\end{enumerate}

\noindent
\textbf{Some problems I added for \S8.4:}
\begin{enumerate}
\item Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
\begin{enumerate}
\item $\displaystyle \sum_{n=0}^\infty \frac{(-10)^n}{n!}$
\item $\displaystyle \sum_{n=1}^\infty (-1)^{n-1}\frac{2^n}{n^4}$
\item $\displaystyle \sum_{n=1}^\infty (-1)^{n}\frac{1}{1+\ln(n)}$
\end{enumerate}

\item Show that $\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{n5^n}$ is convergent. How many terms of the series do we need to add in order to find the sum to within $0.0001$ of its actual value?
\end{enumerate}
\newpage
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%\textbf{\Huge Series Tests Chart}\\
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\noindent
\begin{tabular}{|c|c|c|c|c|}\hline
Name of Test & How it works & Hypotheses & When to use & Example(s)\\\hline&&&&\\[12pt]
Divergence Test &\hspace*{2in} &\hspace*{1.5in} &\hspace*{1.5in}&\hspace*{2in}\\[24pt]\hline&&&&\\[12pt]
Geometric Series &&&&\\[24pt]\hline&&&&\\[12pt]
Integral Test&&&&\\[24pt]\hline&&&&\\[12pt]
Direct Comparison Test&&&&\\[24pt]\hline&&&&\\[12pt]
Limit Comparison Test&&&&\\[24pt]\hline&&&&\\[12pt]
Alternating Series Test&&&&\\[24pt]\hline&&&&\\[12pt]
Ratio Test&&&&\\[24pt]\hline
\end{tabular}
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