\input amssym.def
\def\d{\displaystyle}
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\def\d{\displaystyle}
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\centerline {\largerbold Problems, Week 8}

\bigskip

%10.5, 10.6, 10.7 alternating series. abs. convergence. ratio & root
%tests. some of these questions are (again) quite a bit too clever! 


1.~For each fixed value of $x$, the terms of the series $\displaystyle
{ \sum_{n=1}^{\infty} a_n}$ are given by the recursive formula:
$$\eqalign{a_1 &= x\cr a_{n+1} &= \left( {n \over {n+1}} \right) x^2
a_n\cr}$$ 
Write out the first 5 terms of this series. Guess a general explicit
formula for $a_n$ and discover for which values of $x$ this series
converges. 

\bigskip

2.~For which values of $x$ does the series $\displaystyle {
\sum_{n=1}^{\infty} {{(n!)^3 x^{4n}} \over {(3n)!}}}$ converge? (Be
careful: $3 n!$ and $(3n)!$ and $3 (n+1)!$ and $(3(n+1))!$ are all
wonderfully different!)

\bigskip

3.~a) How  many terms in the following series do you need to add in order
to give an answer within $\d{1\over 10}$ of the true value?
$$\sum_{n=1}^{\infty} {{(-1)^n} \over n}$$

\medskip

b) We will show later that 
$${1 \over e} =  \sum_{n=0}^{\infty} {{(-1)^n} \over {n!}}$$
If you add up the same number of terms of this series as you did in
part a), how close will you be to the true value?
\bigskip

5.~Conditional convergence is {\sl much much} stranger than you think.
\medskip
You have a bag containing an infinite number of 
sticks\footnote*{\sevenrm Don't bother me with reality while I'm
awake, please!}. The $n^{\rm th}$ stick %in the bag 
has length
$\displaystyle{1 \over n}$. You're also given a copy of the
$(x,y)$-plane. An alien named Xwyzq will destroy the Earth
unless $\ldots$ {\it you} can win the following

\centerline{\bf GAME}

{\parindent = .4in \narrower\noindent Xwyzq will specify a point on
the plane, and a positive ``tolerance''.  You will have to select
sticks from your bag and lay them out vertically or horizontally, with
the end of one stick abutting the beginning of the next, and try to
get within the (specified) positive tolerance of the point the alien
has selected in the $(x,y)$-plane.  (You lay your first stick starting
at the origin.) A small picture of one possible path which gets within
$.05$ of the point $(.6,-.3)$ follows: \par }

\centerline{\epsfxsize 1.5in\epsfbox{pic8-1.eps}} 

This is a path of sticks whose endpoint is near the point 
$(.6, -.3)$ (both coordinates of the
endpoint are within $.05$ of what is desired because $\displaystyle{
{1 \over 2} + {1 \over 5} -{1 \over 7} \approx 0.56}$ and because 
$\displaystyle{-{1 \over 4}-{1 \over 8}+{1 \over 10} \approx -0.28}$).  
\bigskip
Of course the question is: 




\centerline{Can {\bf YOU} prevent the destruction of the
Earth?}

That is, can you find a finite selection of sticks to get as
close as Xwyzq specifies to any point in the plane?

After you've thought about this, you might want to contemplate the
answer to the question if the $n^{\rm th}$ stick in the bag
had length $\displaystyle{1 \over {n^2}}$. (Yes, this is a weird
question, but the phenomenon you're investigating is even weirder.)


\bigskip

6.~Consider the series $$ \pm {1 \over 3} \pm {1 \over 9} \pm {1 \over
27} \pm \ldots$$ where {\it you} will choose the signs.
\medskip
a) Is it possible by clever choice of signs to get $0$ as a sum of
this series?
\medskip
b) Is it possible by clever choice of signs to get $\displaystyle{ 1
\over  4}$ as a sum of this series?

\bigskip

7.\footnote*{\sevenrm This series occurs in a text first published
in 1908 by T.J.~I'A.~Bromwich, M.A., Sc.D., F.R.S., based on courses
of lectures given at Queen's College, Galway. A knowledge of
history is valuable for scholars in all fields!}~Suppose $\d
\alpha_n = {1\over {n^2}}$ except when $n$ is a square, and $\d
\alpha_n = {1\over {n^{2/3}}}$ when $n$ is a square.  The series whose
$n^{\rm th}$ term is $\alpha_n$ is thus: $$1 + {1\over {2^2}} +
{1\over {3^2}} + {1 \over {4^{3/2}}} + {1\over {5^2}} + {1\over {6^2}}
+ {1\over {7^2}} + {1\over {8^2}}+ {1 \over {9^{3/2}}}+ \ldots $$ Does
this series converge?

\bigskip
%the following is problem 36 of section 10.6 of the text.

8.~For which positive integers $k$ is the series 
$$\sum_{n=1}^\infty {{(n!)^2} \over {(kn)!}}$$ convergent?
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