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  \textbf{Sample Exam III} \\
  Math 166, Section A2
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  \begin{enumerate}

    \item The series $27-18+12-8+\frac{16}{3}-\frac{32}{9}...$ converges.

      State a reason (with details) for why this series converges (there are two
      different reasons you could give).

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      What value does this series converge to?

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    \item State the integral test; that is, what is it, when can we use it, and what does it tell us?

\vspace*{.4in}
 
      Use the integral test to explain why the harmonic series 
      ${\displaystyle \sum^\infty_{n=1} \frac{1}{n}}$ does not converge.

\vspace*{.2in}

    \item Explain why the alternating harmonic series  
      ${\displaystyle \sum^\infty_{n=1} \frac{(-1)^n}{n}}$ converges.

\vspace*{.2in}

       The first three terms of the series add up to $\displaystyle{\frac{5}{6}}$.  
       Give a bound on how far the exact value of the series is from 
       $\displaystyle{\frac{5}{6}}$.

\vspace*{.2in}

       Why do we say the alternating harmonic series only converges \textit{conditionally}?

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    \item Let us compute the interval of convergence for
      ${\displaystyle \sum^\infty_{n=1} \frac{(-1)^{n+1} x^n}{4^n}}$.

      Use the Ratio Test to determine $r$, the radius of convergence of this series.

\vspace*{.2in}

      Does the series converge at the left endpoint, $x=-r$, of the interval of convergence?

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      At the right endpoint, $x=+r$?

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      Therefore, what is the interval of convergence for this series?

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    \item What is the Taylor Polynomial $p_3(x)$, of degree 3, about the point x=1, that
      approximates $\ln(x)$?

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       Write the formula for the \textit{Taylor remainder}, which measures the difference
       between $p_3(x)$ and $\ln(x)$, at any point $x$.

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    \item Consider the curve whose parametric form is $ x = t^2+4t, y = \sin(2t)$.

      What is a formula for the \textit{horizontal velocity}, $\displaystyle{\frac{dx}{dt}}$? 

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      For $\displaystyle{\frac{dy}{dt}}$?

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      What is a formula for the geometric \textit{slope}, $\displaystyle{\frac{dy}{dx}}$?

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      At what values of $t$ does this curve have vertical tangent lines?

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      At what values of $t$ does this curve have horizontal tangent lines?

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    \item Set up the integral for the arclength, from t=-1 to t=+1, of the curve 
      whose parametric form is $x = t^2, y = 2t.$

\vspace*{.2in}

    \item Consider the curve $r=2+3 \sin(\theta)$.   

      When does this curve have vertical tangent lines? 

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      When does this curve have horizontal tangent lines? 

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      (Your answers may simply be a condition that $\theta$ must satisfy.)

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    \item What is the area contained within the polar curve $r=1-\sin(\theta)$?

  \end{enumerate}

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