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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 12\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}


\centerline
{\bf WORKSHEET 12} 
\bigskip


\item{1.}  State the {\bf Alternating Series Test}.  Give an example
of an alternating series which does not converge.

\medskip 
\item{2.}  Determine whether the following series converge or diverge:
$$
\pt a \sum_{n=2}^\infty {\cos(n\pi)\over \ln n} \qquad
\pt b \sum_{n=1}^\infty (-1)^n{3+(-1)^n\over2n} \qquad
\pt c \sum_{n=2}^\infty (-1)^n{\ln n\over n+1} \hskip1.1in
$$

\medskip
\item{3.}  Find an approximation to the value of 
$$
\sum_{n=1}^\infty(-1)^{n+1}{1\over n}
$$
with an error less than .1.  Is your approximation too large or too small?

\medskip
\itemitem{4.\hskip12pt a)}  Use the first eight terms of the following
series to estimate its actual value:
$$
\sum_{n=0}^\infty (-1)^n{1\over2^n}.
$$

\itemitem{b)}  What can be said about the size of the error of your
approximation in part a)?

\medskip
\itemitem{c)}  Find the {\bf exact} value of this series.  How far off
was your estimate?

\medskip
\item{5.}  Does $\sum a_n$ converge if
$$
\pt a  a_n=\cases{1/3^n,&if $n$ is odd\cr -n/3^n,&if $n$ is even}\qquad
\pt b  a_n={n!\over n^n}\qquad
\pt c  a_1=10,\quad a_{n+1}={a_n\over n}\hskip.4in
$$

\medskip
\item{6.}  Two market women were selling apples, one at 3 apples for a
penny and the other at 2 apples for a penny.  One day when both were
called away they left their stock in charge of a friend.  To simplify
her reckoning the friend amalgamted the stocks - there were 30 apples
of each quality - and sold them all at 5 for two pennies.  Thus the
friend took in 24 pennies.  When it came to dividing the proceeds
between the two owners, trouble arose.  The one who had turned over 30
apples of 3 for a penny quality demanded her due 10 pennies.  The
other not unreasonably asked for 15 pennies.  The sum actually
realized was a penny short.  Where did it (the missing penny) go?







\bye