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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 16\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 16} 
\centerline{\bf a.k.a. Practice Exam 1}


$$
{1.}  \hbox{ Does the series } \sum_{n=2}^\infty(-1)^n{1\over\ln n}
\hbox{ converge absolutely, converge conditionally, or diverge?}
$$



$$
{2.}  \hbox{ Find } \lim_{x\to0}x\ln x.
\hskip4.1in
$$


\medskip
\item{3.}  Consider the Fibonacci sequence which is defined by $F_0=1$,
$F_1=1$, and  
$$
F_{n+2}=F_{n+1}+F_n, \qquad n\geq2.
$$
Find the limit
$$
\lim_{n\to\infty}{F_{n+2}\over F_n}{\atop.}
$$


$$
{4.}  \hbox{ Compute } \int_{-\infty}^0x^2e^x\,dx.
\hskip3.65in
$$


$$
{5.}  \hbox{ Compute }\int{x-1\over x^2+1}\,dx.
\hskip3.65in
$$


$$
{6.}  \hbox{ Find the value of }\sum_{n=1}^\infty
\left({1\over2}\right)^n+\left({1\over3}\right)^n{\atop.}
\hskip2.75in
$$

\medskip
\itemitem{7.\hskip12pt a)}   Find the Taylor series for $f(x)=\ln x$
centered at $x=1$.

\medskip
\itemitem{b)}  What is the interval of convergence for this series?

\medskip
\itemitem{c)}  What is $P_5(x;1)$?


\medskip
$$
{8.}  \hbox{ Prove } \lim_{n\to\infty}{1\over n+2}=0.
\hskip3.69in
$$


$$
{9.}  \hbox{ Find values for $a$ and $b$ such that the power series }
\sum_{n=1}^\infty {a^n(x+b)^n\over n^2}
\hskip1.06in
$$

\item{} converges on $(-3,1)$ and diverges for $x<-3$ and $x>1$.  Then
determine what happens at the endpoints $x=-3$ and $x=1$.


\bigskip
\item{10.}  Find the volume of the solid whose base is a circle of
radius $r$ and with perpendicular cross-sections equilateral triangles.







\bye