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\begin{center}
{\bf APPM 1360 ~~~~~~ FALL 1996}   \\
\medskip

\underline{\bf  Exam 3} \\
November 13, 1996, ~~~  7:30 - 9:00 p.m.
\end{center}

\noindent
ON THE EXAM BOOKLET PLEASE:~~  (1) Write your name,  (2) Student ID,
(3) Lecture Number, and (4) Recitation Instructor.

\underline {Show all work}.  A correct answer with no relevant work may
receive no credit, while an incorrect answer accompanied by some correct
work may receive partial credit.
\begin{center}
\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
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\begin{itemize}

\item [I.] ~ (25 pts.)
  Determine whether the following series converge absolutely, converge
  conditionally, or diverge.  You must show all of your work.
  \begin{itemize}
    \item [A.] ~ 
      $\ds \sum_{n=1}^\infty \frac{1}{n (1 + \ln n)} $
    \item [B.] ~
      $\ds \sum_{n=1}^\infty (-\sqrt{2})^n $
    \item [C.] ~
      $\ds \sum_{n=2}^\infty (-1)^{n+1} \frac{1}{n \ln n} $
  \end{itemize}

\item [II.] ~ (30 pts.)
  Find the interval of convergence for the following series.  Be sure
  to check the endpoints (if there are any) and state whether they
  converge absolutely, conditionally, or diverge. (For two points of
  extra credit each, state the functions to which the series converge.)
  \begin{itemize}
    \item [A.] ~
      $ \ds \sum_{n=0}^\infty (-1)^n \frac{x^{2n + 3}}{(2n+1)!} $
    \item [B.] ~
      $ \ds \sum_{n = 1}^\infty n^n (x - \pi)^n $
    \item [C.] ~
      $ \ds \sum_{n=1}^\infty (1 + \frac{1}{n})^n x^n $
  \end{itemize}

\item [III.] ~ (20 pts.)
  Compute the series for $\ds \tan^{-1}(x)$ by following the steps
  outlined below.
  \begin{itemize}
    \item [A.] ~
      Write down a geometric series which has the following sum $\ds
      \frac{1}{1+x^2}$.
    \item [B.] ~
      Integrate it to find the series for $\ds \tan^{-1}(x)$.  What must
      the constant of integration be?
  \end{itemize}

\item [IV.] ~ (25 pts.)
  Let $a_0 = 1$ and define $a_{n+1} = (\ln x) a_n$.
  \begin{itemize}
    \item [A.] ~
      Write the first four elements of this sequence.  Give the
      formula for the general term of this sequence.
    \item [B.] ~
      For what values of $x$ does the sequence converge?
    \item [C.] ~
      For what values of $x$ does the series $\ds \sum_{n=0}^\infty
      a_n $ converge?  Don't forget to check the endpoints for
      absolute or conditional convergence or divergence.
    \item [D.] ~
      To what function does the series converge?
  \end{itemize}

\end{itemize}

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