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\def\pt#1{\hbox{#1) }}
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\def\R{{\Bbb R}}
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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 8\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 8} 

\bigskip 

\item{1.}   Consider the {\it harmonic series}
$$
1+{1\over2}+{1\over3}+{1\over4}+{1\over5}+{1\over6}+{1\over7}+\cdots
$$
which we have seen is divergent, that is, the partial sums tend to
infinity.  But you might ask at what rate do these partial sums get
large?
\medskip
\itemitem{a)}  Recall that John's computer program added roughly
one billion terms in 30 minutes (to a grand total of 21.3).  Suppose
that you started this program computing the partial sums with $s_1=1$
eleven billion years ago (one estimate of the age of the universe).
Assuming the computer never loses accuracy, how large would $s_n$
be today?  Guess.

\medskip
\itemitem{b)}  Last week we discovered a way to bound the partial sums
of a series (either above or below) by a corresponding integral.
Taking a limit allowed us to test for convergence or divergence of
certain series by looking at what happened to the resulting improper
integral.  We called this the {\bf integral test}.  Use these concepts
to compute upper and lower bounds for the partial sum $s_n$ of the
harmonic series.

\medskip
\itemitem{c)}  Check your guess to part a).

\medskip
\itemitem{d)}  Can you determine how much accuracy John's computer
lost in computing the one billionth partial sum?

\medskip 
\item{2.}  You know that the series $\sum_{n=1}^\infty{1\over n^2}$
converges (Do you remember why?), but to what number does it converge?
This is a difficult, and thus for the purposes of this worksheet
hypothetical, question.  Compute the partial sums out to several
terms.  How close do you think these are to the actual limit?  Be
clever and devise a method to find out how many terms you would need
to add to ensure that your estimate will be off by not more than
$.001$.

\medskip
\itemitem{3.\hskip12pt a)}  Show that ${\lim\atop
c\to\infty}\int_{-c}^cx\,dx$ exists.  {\bf Find it.}

\medskip
\itemitem{b)}  Explain why the limit in a) exists however
$\int_{-\infty}^\infty x\,dx$ diverges.


\eject
\item{4.}  Let $f\colon[0,\infty)\to\R$ be a function with $f(0)=2$
and $\lim_{x\to\infty}f(x)=3$.  Find the following:
$$
\pt a \lim_{b\to\infty}{1\over b}\int_0^bf(x)\,dx \qquad
\pt b \int_0^\infty f(x)\,dx \qquad
\pt c \lim_{b\to0}{1\over b}\int_0^bf(x)\,dx\hskip1in
$$

\medskip
\item{5.}    A working light bulb is in a closed room with no windows.
Outside the room, is a panel of three switches, one of which controls
the light inside (up is on, down is off.)  You may do anything you
like to the three switches and then enter the room to inspect the
light.  After this, without any further experimentation, you must
indicate which switch controls the light.  What do you do?

\medskip
\itemitem{6.\hskip12pt a)}  What is a reasonable value to assign to
the unending expression
$$
\sqrt{2+{\sqrt{2+{\sqrt{2+{\sqrt{2+{\sqrt{2+\cdots}}}}}}}}}
$$
\medskip
\itemitem{b)}  What is a reasonable value to assign to
the unending expression
$$
1+{1\over1+{1\over1+{1\over1+{1\over1+{1\over{1+\cdots}}}}}}
$$




\bye