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\font\largerbold = cmbx10 scaled \magstep2

\centerline {\largerbold Problems, Week 6}

\bigskip

%introduction to sequences and series, covering 10.1 and 10.2. please
%note that once again i have supplied too many problems and perhaps
%some of them are TOO cute or clever. selection of problems will
%obviously depend on exactly what examples are used each lecture. i
%hope people will be able to select appropriate questions.

1.~An infinite sequence of squares is drawn (the first five are
shown), with the midpoints of the sides of one being the vertices of
the next. The outermost square has sides which are 1 unit long. What
is the \underbar{sum} of the \underbar{perimeters} of all of the
squares?

%my first use of pictex in 151-152. thanks to gene speer for this!

\input pictex
\medskip\centerline{
\beginpicture
\setcoordinatesystem units <1truein,1truein>
\putrule from -1 -1 to -1 1
\putrule from -1 1 to 1 1
\putrule from 1 1 to 1 -1
\putrule from 1 -1 to -1 -1
\setlinear  
\plot 0 -1  -1 0  0 1 1 0  0 -1 /
\putrule from -.5 -.5 to -.5 .5
\putrule from -.5 .5 to .5 .5
\putrule from .5 .5 to .5 -.5
\putrule from .5 -.5 to -.5 -.5
\setlinear  
\plot 0 -.5  -.5 0  0 .5 .5 0  0 -.5 /
\putrule from -.25 -.25 to -.25 .25
\putrule from -.25 .25 to .25 .25
\putrule from .25 .25 to .25 -.25
\putrule from .25 -.25 to -.25 -.25
\endpicture}

\bigskip

2.~Determine whether each
of the following sequences converges, and find the limits when they
exist. Explain your answers.
\smallskip
\settabs 2 \columns
\+a) $\displaystyle a_n = {{\ln \, (\ln \, (n^4))} \over {(\ln \,
n)^3}}$ 
&b) $\displaystyle b_n = {\root {n} \of {6n^8}}$\cr

\bigskip

\settabs 3 \columns 
3.~Consider the following sequences:

\smallskip

\+a) $ {a_n = \left( 1 + \left( 1 \over n \right) \right)^n}$
&b) $ {b_n = \left( 1 + \left( 1 \over {n^2} \right) \right)^n}$ 
&c) $ {c_n= \left( 1 + \left( 1 \over {\sqrt n} \right) \right)^n}$\cr

\smallskip 

Find the first five terms of each of these sequences. What can you say
about the convergence of each of these sequences? (Why?)

\bigskip

\settabs 2 \columns

4.~Suppose $\alpha$ and $\beta$ are positive numbers. Decide whether
the following sequences converge. If they converge, try to find their
limits.\footnote\dag{\sevenrm Your answers may involve both numbers
and their relationship.} (You must give reasons for your assertions!)

\smallskip

\+a) $d_n= \root n \of {\alpha^n + \beta^n}$
&b) $e_n= \root n \of {\alpha^n + \beta}$\cr

\bigskip

5.~a) Compute the first five terms of the sequence ${
\left\{ n \arctan \left( 1 \over n \right) \right\} }$. Do you think
this sequence converges? If you do, what do you think the limit is?

\medskip

b) Rewrite $ { n \arctan \left( 1 \over n \right) }$ as
$\displaystyle{ { \arctan \left( 1 \over n \right) } \over { 1 \over
n} }$.  Then use l'Hospital's rule to answer the questions asked in
part a).

\medskip

c) Redo a) and b) for the sequence $ { \left\{ n \left( \sqrt{ 1+ {3
\over n} } - 1 \right) \right\} }$.

\medskip

d) Suppose $f$ is a differentiable function and you know that
$f(0)=0$.  Then conclude that the sequence $ { \left\{ n \, f\!\left(
1 \over n \right) \right\} }$ converges, and tell what its limit is.

\bigskip

6.~Consider the differential equation $$y'' = - y$$ with initial
conditions $y(0)=0$ and $y'(0)= 1$ (this is what is called the
``initial velocity'' solution). Suppose that the solution looks like
$$y = A + Bx + Cx^2 + Dx^3 + Ex^4 + Fx^5 +G x^6 + Hx^7 + \ldots$$ where
the letters $A, B, C,
\ldots$ are used to represent unknown constants. Use the initial
conditions to find values of some of these constants. Then use the
differential equation to deduce values of the other constants. Can you
see a general pattern? What is the pattern?
\medskip
$\underline{\rm Special\ added\ bonus\ question}$: what well-known
function $\underline{\rm is}$
the ``initial velocity'' solution? What do you think will be the relation
between the series you have discovered and this function?

\bigskip

7.~a) Two students are sharing a loaf of bread. Student Alpha
eats half of the loaf, then student Beta eats half of what's left,
then Alpha eats half of what's left, and so on. How much of the loaf
will each student eat?

\medskip 

b) Two students are sharing a loaf of bread. Student Alpha, now
hungrier and more ferocious, eats two-thirds of the loaf, then student
Beta eats eats half of what's left, then Alpha eats two-thirds of
what's left, then Beta eats half of what's left, and so on. How much
of the loaf will each student eat?

\medskip 

c) Now let's start with three students: Alpha, Beta, and Gamma. They
decide to share a loaf of bread. Alpha eats half of the loaf, passes
what's left on to Beta who eats half, and then on to Gamma who eats
half, and then back to Alpha who eats half, and so on. How much of the
loaf will each student eat?

\bigskip

8.~a) If you enter 5 in your 10 digit calculator, and then hit the
square root button 20 times, you'll probably get: $$1.00000\, 15348$$
and if you then subtract 1 and multiply by 1,048,576 you'll get:
$$1.60943\, 91475$$ but on the other hand, the same calculator will tell
you that $\ln 5$ is: $$1.60943\, 79124$$ and I wonder, is this just a
coincidence? {\bf Well, is it?} Hint: 1,048,576 is $2^{20}$.
\medskip

b) Given a positive number, $x$, outline a strategy for computing $\ln
x$ only with the ``primitive'' arithmetic operations ($+$, $\times$,
$-$, $/$) and square root (\raise
.2em\hbox{$\sqrt{\hphantom{zi}}$}). Your strategy should involve
asserting (and verifying) that a certain sequence which can be easily
computed with the listed operations always converges to $\ln x$.

\bigskip

9.~{\sevenrm Bounded increasing sequences converge.}\par
a) Prove: if $0\le w \le 87$, then $0\le \sqrt{3w+ 42} \le 87$.

\medskip

b) Prove: if $0\le u \le v \le 87$, then $0 \le \sqrt{3u+42} \le
\sqrt{3v+42} \le 87$. 

\medskip

c) Consider the following recursively defined sequence: $$\eqalign{v_1
&= 1\cr v_{n+1} &= \sqrt{3v_n + 42}\cr}$$ and compute its first five
terms. Prove that this sequence converges.\par {\sevenrm It sure is
interesting that bounded increasing sequences converge. See p.585 of
the text.}

\medskip

d) Suppose $V$ is the limit of the sequence defined in c). What is
$V$? (Square the recursive equation and take limits using some limit
theorems.)

Note: this process is used in many, many sequences
encountered in ``real life''.\par  {\sevenrm Oh, by the way, bounded
increasing sequences converge.}

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