\def\tT{Math126, Test III}
\def\ddd{April 20, 1999}
\def\rightanswers{
\ll 1 4
\ll 2 2
\ll 3 1
\ll 4 5
\ll 5 5
\ll 6 3
\ll 7 4
\ll 8 1
\ll 9 5
\ll{10} 2
}

\def\isum{\displaystyle\sum_{n=1}^{\infty}}
\input 126Header.tex
\exam{
\centerline{\bf Multiple Choice}


\pr(5pts) Determine whether the following series converge
or diverge.
$$1) \sum_{n=1}^{\infty} \frac{(-1)^n}{n}, \;\;\;\;\;\;
2) \sum_{n=1}^{\infty} \frac{1}{(\ln n)^n},\;\;\;\;\;\;
3) \sum_{n=2}^{\infty} \frac{\sqrt{n^3-1}}{3n -1}.$$

\fivelines
{1) 2) and 3) converge}
{1) absolutely converges, 2) and 3) diverge}
{1) conditionally converges, 2) and 3) diverge}
{1) conditionally converges, 2) absolutely converges and 3) diverge}
{1) 2) and 3) diverge} %% (d)


\def\stick{\vrule height0pt depth 10pt width 0pt}

\problemskip\pr(5pts)
Find the radius $R$ of convergence of the following power series
$$\sum_{n=1}^{\infty} \frac{(x-5)^n}{n^n}.$$
\ans{$R=0$}{$R=\infty$}{$R=1$}{$R=5$}{$R=\sqrt{5}$} %% (b)

\newpage\pr(5pts)
Use the definition to find the Maclaurin
series for the
function
$$\frac{1}{(1-2x)^2}$$
\threetwo{$\sum_{n=1}^{\infty} (-1)^n n 2^{n} x^{n-1}$}
{$\sum_{n=1}^{\infty} (-1)^n n 2^{(n+1)} x^{n}$}
{$\sum_{n=1}^{\infty} (-1)^n n 2^{(n-1)} x^{n-1}$}
{$\sum_{n=1}^{\infty} (-1)^n  2^{n} x^{n}$}
{$\sum_{n=1}^{\infty} (-1)^n (n+1) 2^{(n-1)} x^{n-1}$}
%%(a)

\problemskip\pr(5pts)
Give the Maclaurin series of the function
$f(x) = \sin(x^2).$

\threetwo{$\sum_{k=0}^{\infty} \frac{(-1)^{(2k+1)}}{(2k+1)!} x^{4k+2}$}
{$\sum_{k=1}^{\infty} \frac{(-1)^k}{(2k+1)!} x^{4k}$}
{$\sum_{k=0}^{\infty} \frac{(-1)^{(k+1)}}{(2k+1)!} x^{2k+2}$}
{$\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)!} x^{2k+1}$}
{$\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)!} x^{4k+2}$}
%%(e)

\newpage\pr(5pts)
Give the first three nonzero terms of the Maclaurin  series
expansion of $e^x \sin x$.
\threetwo{$x+x^2+\frac{2}{3}x^3$}{$x-x^2-\frac{1}{2}x^3$}
{$x+3x^2+\frac{1}{6}x^3$}{$x+x^2+\frac{1}{3}x^3$}
{$x+x^2-x^3$} %%(d)

\problemskip\pr(5pts)
%%%Find, if it exists,
%%%$$\lim_{n\to\infty} \ln n - \ln (n+1).$$
%%\ans{$1$}{diverges}{$0$}{$e$}{$e^{-1}$} %% (c)
Which series conditionally converges?
\ans{$\isum \frac1{n^2}$}
{$\isum \frac1{n}$}
{$\isum \frac{(-1)^{n+1}}{n}$}
{$\isum \frac{(-1)^{n+1}}{n^2}$}
{$\isum \frac1{n^2+n}$}

\newpage\pr(5pts)
Find the sum of the following series
$$\sum_{n=0}^{\infty} e^{-n} - e^{-(n+1)}.  $$
\ans{diverges}{$e$}{$e^{-1}$}{$1$}{$2$} %% (d)


\problemskip\pr(5pts)
Determine the set of all values $x$ such that the  series
$$\sum_{n=0}^{\infty}(\ln x)^n  $$
converges.
\ans{$e^{-1}<x<e$}{diverges for all $x$}{$1<x<e$}{$e^{-1}\leq x<0$}
{converges for all $x$} %% (a)
\newpage\pr(5pts)
Compute
$$\lim_{x\to\infty}\frac{2(\cos(x)-1)+2x^2}{x^4}$$
\ans{$+\infty$}{$0$}{$\frac{1}{24}$}{$1$}{$\frac{1}{12}$} %% (e)

\problemskip\pr(5pts)
Find the third order term of the Maclaurin expansion of the solution $y(x)$
of the following initial value problem:
$$y'=y+x^2, \,\,\,\,\,\,\, y(0)=-2$$
\ans{$x^3$}{$0$}{$\frac{x^3}{3}$}{$3x^3$}{$\frac{x^3}{3!}$} %% (b)

\newpage\centerline{\bf Partial Credit}

\pr(10pts)
Find interval of convergence of the following series
$$\sum_{n=1}^{\infty} (-1)^{n+1}
\frac{(1-x)^n}{n}.$$

\newpage\pr(10pts)
Find Maclaurin series representation for the function
$$\int_0^x e^{-t^2} dt.$$

\newpage\pr(10pts)
Find interval of convergence of the following power series:
$$\sum_{n=1}^{\infty}  \frac{(2n)^2}{3}(x-2)^{n}.$$
Be sure to
investigate the endpoints of the interval.

\newpage\pr(10pts)
If $0<x<0.5$,  use Alternating Series Theorem and the Binomial Theorem to
show that
$\sqrt{1+x}\approx 1+ 0.5 x$ with  an error less than $0.032$. Notice that the
series is alternating after the first term.

\newpage\pr(10pts)
Find 4 first terms of the Taylor series for $f(x)=\cos x$ about $a=\pi$.
\newpage
}



\print{Bullwinkle}
%\print{Cao}
%\print{Alber}
%\print{Taylor}
%\print{Nollet}
%\print{Patrizio \quad{\sl Section \#:}}
%\marked{}
\end

