 
\begin{center} 
{\large \bf PROBLEM SET 20.1}
\end{center}
\vspace{.2in}

\noindent 
$ ~~ \: ${\bf 1.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
Compute the first 5 terms of each of the sequences
$ \{ a_n \} $.
\vspace{.1in}

\begin{tabular}{ll}
(a) $ a_n = \sin \: \frac{ n \pi }{ 2 } $ &
(b) $ a_n = \frac{ 1 }{ 2 } [ ( 1 + \sqrt{ 5 } )^n + ( 1 - \sqrt{ 5 }^n ] $ 
\\ [0.2in]
(c) $ a_n = 2^{ \cos \: n \: \pi / 3 } $ & 
\end{tabular}
\end{minipage}
\vspace{.3in}

\noindent
{\bf Problems 3--15 give the nth term of a sequence.
Find the limit if it exists.  If the sequence diverges,
say so.  Use Theorem 1, 2, and 3, the techniques
introduced in the Examples, and known properties of
functions.} 
\vspace{.2in}

\noindent  
\begin{tabular}{lll}
$ ~ \: ${\bf 3.} $ ~~ $
$ a_n = {\displaystyle \frac{ n^2 + 1 }{ n^2 }} $
& \hspace{1in} &
$ ~ \: ${\bf 4.} $ ~~ $ 
$ a_n = {\displaystyle \frac{ 3n^4 - 5n^3
+ 7n - 1 }{ 5n^4 + 4n^3 - 8n + 15 }} $
\\ [0.2in]
$ ~ \:${\bf 5.} $ ~~ $ 
$ a_n = e^{ 1/n } $
&& $ ~ \:${\bf 6.} $ ~~ $ 
$ a_n = ( 0.7 )^n $
\\ [0.2in]
$ ~ \:${\bf 7.} $ ~~ $ 
$ a_n = ( 1.01 )^n $
&& 
$ ~ \:${\bf 8.} $ ~~ $ 
$ a_n = {\displaystyle \frac{ \ln \: n )^2 }{ n }} $
\\ [0.2in]
$ ~ \:${\bf 9.} $ ~~ $ 
$ a_n = {\displaystyle \int_{ - 1/n }^{ 1/n }} \:
\sqrt{ 1 + t^4 } \: d t $
&& {\bf 10.} $ ~~ $ 
$ a_n = {\displaystyle \int_{ - n }^n } \: 
{\displaystyle \frac{ d t }{ 1 + t^2 }} $ 
\\ [0.2in]
{\bf 11.} $ ~~ $ 
$ a_n = \left ( 1 + \frac{ 3 }{ n } \right )^n $
&& {\bf 12.} $ ~~ $ 
$ a_n = ( n + 1 )^{ 1/n } $
\\ [0.2in]
{\bf 13.} $ ~~ $ 
$ a_n = \left ( 1 - \frac{ 1 }{ n^2 } \right )^{ 1/n } $
&& 
\\ [0.2in]
{\bf 14.} $ ~~ $ 
$ a_n = [ \sin \: 1/n ]^{ 1/n } $
&& \hspace{-1.2in} [Hint:  
$ \frac{ 2 x }{ \pi } < \sin \: x < x $ for $ 0 < x < \pi / 2 $.] 
\\ [0.2in]
{\bf 15.} $ ~~ $ 
$ a_n = \frac{ 4^n }{ n ! } $
\end{tabular} 
\vspace{.2in}

\noindent 
{\bf 16.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
Does the convergence of $ \{ | a_n | \} $
always imply the convergence of $ \{ a_n \} $?
\end{minipage}
\vspace{.2in}

\noindent 
{\bf 17.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
If $ a_n \rightarrow 3 $ and $ b_n \rightarrow 4 $
as $ n \rightarrow \infty $, what about the sequence 
$ a_1 , b_1 , a_2 , b_2 , a_3 , b_3 ,... $?
\end{minipage}
\vspace{.2in}

\newpage

\begin{center}
{\bf CALCULATOR PROBLEM}
\end{center}
\vspace{.1in}

\noindent 
{\bf 18.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
(a) The sequence $ a_n = 50^{ 1/n } $
has limit 1.  Compute a table comparing
$ a_n - 1 $ to $ \frac{ 1 }{ n } \: \ln \: 50 $
for the values $ n = 50 , ~ 100 , ~
1000 ,  ~ 2000 $ and observe that, of the
two, $ a_n - 1 $ is lightly larger. 
\\ [0.2in]
(b) Apply the Mean Value Theorem for
derivatives to $ 50^x 
= e^{ x \: \ln \: 50 } $ to conclude that
\[50^{ 1/n } - 1 = e^{ \theta \: \ln \: 50 }
\cdot \frac{ 1 }{ n } \: \ln \: 50 \]
for some $ \theta $ between 0 and $ 1/n $. 
\\ [0.2in]
(c) Use the estimates $ e < 3 $ and $ \ln \: 50 
< 4 $ to conclude that $ 50^{ 1/n } - 1 
< 12/n $ if $ n > 4 $.
\end{minipage}
\vspace{.2in}

 
\newpage

\begin{center} 
{\large \bf PROBLEM SET 20.2}
\end{center}
\vspace{.2in}

\noindent
{\bf In Problems 1--5, decide whether the sequence 
is eventually increasing, eventually decreasing, 
or neither.}
\vspace{.2in}

\noindent  
\begin{tabular}{lll}
$ ~ \: ${\bf 1.} $ ~~ $
$ a_n = {\displaystyle \frac{ 1 }{ 1 + n^2 }} $ 
& \hspace{1in} &
$ ~ \: ${\bf 2.} $ ~~ $ 
$ a_n = {\displaystyle \frac{ 2^n }{ n ! }} $ 
\\ [0.2in]
$ ~ \: ${\bf 3.}
$ a_n = {\displaystyle \frac{ 10^n }{ n ! }} $ 
&&
$ ~ \: ${\bf 4.} $ ~~ $ 
$ a_n = {\displaystyle \int_{ n }^{ n + 1 }} \: 
{\displaystyle \frac{ d x }{ 1 + x^4 }} $ 
\\ [0.2in]
$ ~ \: ${\bf 5.} $ ~~ $ 
$ a_n = {\displaystyle \frac{ 1 }{ n }}
- \ln \: {\displaystyle \frac{ n + 1 }{ n }} $ &&  
\end{tabular}
\vspace{.2in}

\noindent 
{\bf Use Theorem 5 to show that the next four sequences are convergent.}
\vspace{.2in}

\noindent 
\begin{tabular}{lll} 
$ ~ \: ${\bf 6.} $ ~~ $ 
$ a_n = {\displaystyle \frac{ 2 \cdot 4 \cdot 6 
\cdots 2n }{ 1 \cdot 3 \cdot 5 \cdots ( 2 n + 1 ) }} $ 
& \hspace{.6in} & 
$ ~ \: ${\bf 7.} $ ~~ $ 
$ a_n = {\displaystyle \frac{ ( n ! )^2 }{ ( 2 n ) ! }} $ 
\\ [0.2in]
$ ~ \: ${\bf 8.} $ ~~ $ 
$ a_n = {\displaystyle \int_0^n } \:
e^{ - x^2 } \: d x $
&& $ ~ \: ${\bf 9.} $ ~~ $ 
$ a_n = {\displaystyle \frac{ 1 }{ ( n + 1 )^2 }
+ \frac{ 1 }{ ( n + 2 )^2 } + \cdots + 
\frac{ 1 }{ ( 2 n )^2 }} $
\end{tabular}
\vspace{.2in}

\begin{center}
{\bf CALCULATOR PROBLEMS}
\end{center}

\noindent 
{\bf 10.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
Choose $ x_1 $ positive and define 
the sequence $ \{ x_n \} $ by
\[ x_{ n + 1 } = 
\frac{ 5 }{ 1 + x_n } . \]
(a) Construct tables for various choices of $ x_1 $
to show that the terms seem to lie alternately to the left
and to the right of $ L = 1.7912878... $ 
\\ [0.2in] 
(b) Argue geometrically that the terms $ a_n $
oscillate about the limit $L$ with the even terms forming an 
increasing sequence and the odd terms
forming a decreasing sequence.  [Look at the graphs
$ y = x $ and $ y = \frac{ 5 }{ 1 + x } $.]
\\ [0.2in] 
(c) Show that $L$ is the positive root of the quadratic
$ x^2 + x - 5 = 0 $.
\end{minipage}
\vspace{.2in}

\noindent 
{\bf 11.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
Compute the nth term of the sequence
\[ a_n = \frac{ 1 }{ n + 1 } 
+ \frac{ 1 }{ n + 2 } + \cdots + 
\frac{ 1 }{ 2n } . \]
for $ n = 25 , ~ 50 , ~ 100 , ~ 200 $ and show that
\[ \ln \: \frac{ 2n + 1 }{ n + 1 }
< a_n < \ln \: 2 . \]
\end{minipage}

\newpage 
 
\begin{center} 
{\large \bf PROBLEM SET 20.3}
\end{center}
\vspace{.2in}

\noindent
{\bf In Problems 1--7, decide whether the series converges
or diverges.  If it diverges, state why.  If
it converges, find its sum.}
\vspace{.2in}

\noindent  
\begin{tabular}{ll}
$ ~ \: ${\bf 1.} $ ~~ $
$ 1 + \frac{1}{2} + \frac{1}{2^2} 
- \frac{1}{2^3} + \cdots $ 
& \hspace{-.5in} $ ~ \: ${\bf 2.} $ ~~ $ 
$ 1 + e^{ - 1 } + 
e^{ - 2 } + e^{ - 3 } + \cdots $
\\ [0.2in]
$ ~ \: ${\bf 3.} $ ~~ $ 
$ 1 + 0 + 
\frac{1}{2} + 0 + 
\frac{ 1 }{ 3 } + 0 + 
\frac{ 1 }{ 4 } + \cdots $
& \hspace{-.5in}
$ ~ \: ${\bf 4.}
$ {\displaystyle \sum_{n=0}^{ \infty }} \: 
( - 1 )^{ n + 2 } ( .9 )^n $
\\ [0.2in]
$ ~ \: ${\bf 5.} $ ~~ $ 
$ {\displaystyle \sum_{n=0}^{ \infty }} \: 
\frac{ 1 }{ R } \left ( \frac{ r }{ R } \right )^n ~~
( 0 < r < R ) $
& \hspace{-.5in} 
$ ~ \:${\bf 6.} $ ~~ $ 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: 
\ln \: \left ( \frac{ 3n + 5 }{ 2n + 1 } \right ) $
\\ [0.2in]
$ ~ \:${\bf 7.} $ ~~ $ 
$ {\displaystyle \sum_{n=0}^{ \infty }} \: 
\frac{ e^{ - k s } }{ s^k } ~~ ( s = ~ \mbox{positive constant}) $
& \end{tabular}
\vspace{.2in}

\noindent 
$ ~~ \: ${\bf 8.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
Use the geometric series formula
and (10) to find the rational
representation for each of the following
infinite decimals. \\ [0.2in]
\begin{tabular}{lll} 
(a) $ .494949... $ & \hspace{2in} & 
(b) $ .123123...$
\\ [0.1in]
(e) $ .ababab... $ && \hspace{-1.8in}
($a$ and $b$ one of $ 0,1,2,...,9$) \end{tabular} 
\end{minipage}
\vspace{.2in}

\noindent 
$ ~~ \:${\bf 9.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
Replace $x$ by $ x^2 $
in the geometric series (6) to find a series representation 
for $ \frac{ 1 }{ 1 - x^2 } $.  Then obtain 
the same series another way:
add the series $ 1 + x + x^2 + \cdots $ and 
$ 1 - x + x^2 - \cdots $.
\end{minipage}
\vspace{.2in}

\noindent 
{\bf 10.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
Use the geometric series and (10) to find the sum of each
of the following series.  Assume $ | x |
< 1 $. \\ [0.2in]
\begin{tabular}{lll} 
(a) $ x^2 + x^3 + x^4 + \cdots $
& \hspace{2in} & 
(b) $ 1 + \sin \: \frac{ \pi x }{ 2 } + 
\sin ^2 \frac{ \pi x }{ 2 } + \cdots $
\end{tabular}
\end{minipage}
\vspace{.2in}

\noindent 
{\bf 11.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
Choose $ a > 0 $ and define the function $f$ by
\[ f(x) = \frac{ 1 }{ a } 
- \frac{ 1 }{ a }
\left ( \frac{ ( x - a ) }{ a } \right )
+ \frac{ 1 }{ a }
\left ( \frac{ ( x - a ) }{ a } \right )^2
- \frac{ 1 }{ a }
\left ( \frac{ ( x - a ) }{ a } \right )^3 + \cdots \]
Find the domain of $ f(x) $ and a simple
formula for $ f(x) $ in that domain.
\end{minipage}
\vspace{.2in}

\noindent 
{\bf 12.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
Find the terms of the infinite series having partial sums
\[ s_n = \frac{1}{2} + \frac{1}{2}
\: {\displaystyle \frac{ \sin \: \left ( n + \frac{1}{2} 
\right ) \theta }{ \sin \: \frac{ \theta }{ 2 } }} , ~~~
n = 0 , 1 , 2 ,... ~~ . \]
\end{minipage}

 \newpage

\begin{center} 
{\large \bf PROBLEM SET 20.4}
\end{center}
\vspace{.2in}

\noindent
{\bf In Problems 1--6, 
show that the Integral Test can be applied and then
use it to determine whether the series converges
or diverges.}
\vspace{.2in}

\noindent  
\begin{tabular}{lll}
$ ~ \: ${\bf 1.} $ ~~ $
$ {\displaystyle \sum_{n=1}^{ \infty }} \: 
\frac{ 1 }{ 5n^2 + 1 } $
& \hspace{2in} & 
$ ~ \: ${\bf 2.} $ ~~ $ 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: 
\frac{ 1 }{ ( 3n + 1 )^2 } $
\\ [0.2in]
$ ~ \: ${\bf 3.}
$ {\displaystyle \sum_{n=1}^{ \infty }} \: \frac{ 2n + 1 }{ n^2
+ n + 1 } $
&& $ ~ \:${\bf 4.} $ ~~ $ 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: n e^{ - n } $
\\ [0.2in]
$ ~ \:${\bf 5.} $ ~~ $
$ {\displaystyle \sum_{n=1}^{ \infty }} \: \frac{ \tan ^{- 1}
n }{ 1 + n^2 } $
&& $ ~ \:${\bf 6.} $ ~~ $ 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: 
\frac{ 1 }{ \sqrt{ n } + 5 } $
\end{tabular}
\vspace{.2in}

\noindent 
{\bf In the next seven problems establish the convergence
or divergence of each series by application 
of one of the comparison tests.}
\vspace{.2in}

\noindent 
\begin{tabular}{lll} 
$ ~ \: ${\bf 7.} $ ~~ $ 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: \frac{ 1 }{ 10
n^2 + 17 } $ & \hspace{2in} & 
$ ~ \: ${\bf 8.} $ ~~ $ 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: 
\frac{ 3^n + 2 }{ 4^n + 3 } $
\\ [0.2in]
$ ~ \: ${\bf 9.} $ ~~ $ 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: 
\frac{ 1 }{ n ( n^2 + n + 1 ) } $
&& {\bf 10.} $ ~~ $ 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: \frac{ 1 }{ n ( n^2 + n + 1 ) } $
\\ [0.2in]
{\bf 11.} $ ~~ $ 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: \frac{ 1 }{ \sqrt{ n^2 + n + 1 } } $
&& {\bf 12.} $ ~~ $ 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: 
\frac{ \sqrt{ n }}{ n^2 + e^{ - k }} $
\end{tabular}
\vspace{.2in}

\noindent  
$ ~ \: ${\bf 13.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
Use the Mean Value Theorem to show that
$ 0 < x - \ln \: ( 1 + x ) < x^2 $ if $ x > 0 $,
and hence that the series
$ {\displaystyle \sum_{n=1}^{ \infty }} \: 
\left ( \frac{ 1 }{ n } - 
\ln \: \left ( 1 + \frac{ 1 }{ n } \right ) \right ) $
converges while both $ {\displaystyle \sum_{n=1}^{ \infty }}
\: \frac{ 1 }{ n } $
and $ {\displaystyle \sum_{n=1}^{ \infty }} \: 
\ln \: \left ( 1 + \frac{ 1 }{ n } \right ) $ are
divergent series.
\end{minipage}
\vspace{.2in}

\noindent 
{\bf In the next two problems, first use the Mean Value
Theorem to estimate the nth term;
then determine convergence or divergence by one
of the comparison tests.}
\vspace{.2in}

\noindent  
\begin{tabular}{lll}
{\bf 14.} $ ~~~ $
$ {\displaystyle \sum_{n=1}^{ \infty }} \: \sin ^2 
\frac{ 1 }{ n } $
& \hspace{1.5in} & 
{\bf 15.} $ ~~~ $
$ {\displaystyle \sum_{n=1}^{ \infty }} \: ( e^{ 1/n } - 1 ) $
\end{tabular}
\vspace{.2in}

\noindent 
$ ~ \: ${\bf 16.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
From the inequality $ {\displaystyle \int_{ 1 }^{ 1 
+ 1/n }} \: \frac{ d x }{ n } > 
\frac{ 1 }{ 1 + n } $ and by comparing areas,
show that  
\[ s_n = 1 = \frac{1}{2} + \frac{1}{3} 
+ \cdots + + \frac{1}{n} - \ln \: n \] 
is a decreasing sequence of positive numbers.
\end{minipage}
\vspace{.2in}

 

\begin{center}
{\bf CALCULATOR PROBLEMS}
\end{center}
\vskip -2ex

\noindent 
{\bf In Problems 17 and 18, find an index 
$ \boldmath \mbox{$N$} $ for which the partial 
sum $ \boldmath \mbox{$s_N$} $
approximates the sum $ \boldmath \mbox{$S$} $
of the given convergent series with an error
less than the indicated value $ \boldmath \mbox{$ \epsilon $} $.
Then compute $ \boldmath \mbox{$s_N$} $
and express the results in the form 
$ \boldmath \mbox{$s_N < S < s_N + \epsilon $} $.
Follow the procedure outlined in Example 2.}
\vspace{.2in}

\noindent 
\begin{tabular}{lll} 
{\bf 17.} $ ~~~ $
$ {\displaystyle \sum_{n=1}^{ \infty }} \: \frac{ 1 }{ n^4 } ,
~~ \epsilon = .00005 $ 
& \hspace{1.5in} 
{\bf 18.} $ ~~~ $
$ {\displaystyle \sum_{n=2}^{ \infty }} \: \frac{1}{ n
( \ln \: n )^2 } , ~~ \epsilon = .1 $
\end{tabular} 


\newpage


\begin{center} 
{\large \bf PROBLEM SET 20.5}
\end{center}
\vspace{.2in}

\noindent
{\bf In Problems 1--9, 
determine whether the given series is convergent or divergent.}
\vspace{.2in}

\noindent  
\begin{tabular}{lll}
$ ~ \: ${\bf 1.} $ ~~ $
$ 1 
- {\displaystyle \frac{ 1 }{ \sqrt{ 2 } } 
+ \frac{ 1 }{ \sqrt{ 3 } } - \frac{ 1 }{ \sqrt{ 4 } } } + \cdots $
& \hspace{1in} & 
$ ~ \: ${\bf 2.} $ ~~ $ 
$ 1 - e^{ - 2 } + 
e^{ - 3 } - e^{ - 4j } + \cdots $
\\ [0.2in]
$ ~ \: ${\bf 3.}
$ {\displaystyle \sum_{ n = 1 }^{ \infty }} \:
( - 1 )^{ n - 1 } {\displaystyle \frac{ 1 }{ 4n^2 }} $
&& $ ~ \:${\bf 4.} $ ~~ $ 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: (-1)^{ n - 1 }
( \ln \: n )^{ - 2 } $
\\ [0.2in]
$ ~ \:${\bf 5.} $ ~~ $
$ {\displaystyle \sum_{n=1}^{ \infty }} \: 
(-1) ^{ n - 1 } \left ( {\displaystyle \frac{1}{2} 
- \frac{ 1 }{ n }} \right )^{ - n } $
&& $ ~ \:${\bf 6.} $ ~~ $ 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: 
(-1) ^{ n - 1 } {\displaystyle \frac{ n^2 }{ n^3 + n }} $
\\ [0.2in]
$ ~ \:${\bf 7.} $ ~~ $
$ {\displaystyle \sum_{n=0}^{ \infty }} \: 
{\displaystyle \frac{ ( - 2 )^n }{ n ! }} $
&& $ ~ \:${\bf 8.} $ ~~ $ 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: (-1)^{ n - 1 }
{\displaystyle \frac{ n^2 }{ n^2 + n }} $
\\ [0.2in]
$ ~ \:${\bf 9.} $ ~~ $
$ {\displaystyle \sum_{n=1}^{ \infty }} \: 
(-1) ^{ n - 1 } n e^{ - n } $
&& 
\end{tabular}
\vspace{.2in}

\noindent 
{\bf In Problems 10--12 indicate the values of $ \boldmath
\mbox{$x$} $ for which the given series alternates in sign.}
\vspace{.2in}

\noindent 
\begin{tabular}{lll} 
{\bf 10.} $ ~~ $ 
$ 1 - {\displaystyle \frac{ x^2 }{ 2 ! }}
+ {\displaystyle \frac{ x^4 }{ 4 ! }} - 
{\displaystyle \frac{ x^6 }{ 6 ! }}
+ \cdots $
& \hspace{.5in} & 
{\bf 11.} $ ~~ $ 
$ x - {\displaystyle \frac{ x^3 }{ 3 ! }}
+ {\displaystyle \frac{ x^5 }{ 5 ! }}
- {\displaystyle \frac{ x^7 }{ 7 ! }} + \cdots $
\\ [0.2in]
{\bf 12.} $ ~~ $ 
$ ( x - 1 )^2 - ( x - 1 )^3 + ( x - 1 )^4 - \cdots $
&& \end{tabular}
\vspace{.2in}

\noindent  
$ ~ \: ${\bf 13.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: 
(-1)^{ n - 1 } \: {\displaystyle \frac{ 1 }{ n ! }} ~~ $, 
$ ~~~ $ error $ < .0001 $
\end{minipage}
\vspace{.2in}

\noindent  
$ ~ \: ${\bf 14.} $ ~~~ $
\begin{minipage}[t]{5.9in} 
$ {\displaystyle \sum_{n=1}^{ \infty }} \: 
(-1)^{ n - 1 } {\displaystyle \frac{ 1 }{ n 
( n + 1 ) }} ~~ $, $ ~~~ $ error $ < .01 $
\end{minipage}




