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\title{Math 399, Homework \#6}


\author{Marius Ionescu}


\date{Due Friday, 03/11/2011}
\maketitle

From the Sierpinski gasket notes: problems 3.1.2 (on page 68), 3.2.4
and 3.2.8 (on page 72).

\setcounter{thm}{3}
\begin{problem}%7
Prove that if the terms of the Harmonic Series containing a 7
in (the base-10 numerals for) their denominator are deleted, the
resulting series converges.
\end{problem}

\begin{problem}%11
Find the Fourier series representation of the saw-toothed
function $f(x)$ defined by:
$$
f(x) = \left\{\begin{array}{l@{\ \rm for\ }l}
   x + \pi & x\in(-\pi,0] \\ \pi - x & x \in (0,\pi] \end{array}
\right.\ ,
$$
and $f(x + 2\pi) = f(x)$ for all $x$.  Use your answer to evaluate
$\sum_{n=1}^\infty \frac1{n^2}$.
\end{problem}

\begin{problem}%29
A little boy has his dollar changed into 6 coins.
Running down the street he drops one coin into a sewer
grate. What is the probability that it was a dime?
\end{problem}

\begin{problem}%36
Let $S$ be a set of 52 distinct positive integers.
Prove that there exist $x, y \in S$ such that either $x + y$
or $|x - y|$ is divisible by 100.
\end{problem}

\begin{problem}%P05
  Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a \rfloor,
\lfloor 2a \rfloor) = 0$ for all real numbers $a$.
(Note: $\lfloor \nu \rfloor$ is the greatest integer less than
or equal to $\nu$.)
\end{problem}
\end{document}
