\documentclass{article}

\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amsmath}

\begin{document}

\theoremstyle{definition}
\newtheorem{problem}{Problem}

\title{Math 2001 Assignment 14}
\date{\today}
\author{Jonathan Wise}
\maketitle

\begin{problem}
Write an English definition meaning ``$x$ is a perfect square''.  Then write a symbolic sentence meaning the same thing, using only quantifiers ($\forall$, $\exists$), variables, set membership ($\in$), logical connectives ($\wedge$, $\vee$, $\neg$, $\Rightarrow$, $\Leftrightarrow$), the set of integers ($\mathbb{Z}$), arithmetic (addition, subtraction, multiplication, division, exponentiation), and equality ($=$).
\end{problem}

\begin{problem}
Use summation notation to rewrite the following expression without ellipses:
\begin{equation*}
1 + \frac{1}{4} + \frac{1}{9} + \cdots + \frac{1}{n^2}
\end{equation*}
\end{problem}

\begin{problem}
Compute $\frac{2001!}{1999!}$ and justify your answer.
\end{problem}

\begin{problem}
Use summation notation to rewrite the following expression without ellipses:
\begin{equation*}
n^2 = 1 + 3 + 5 + 7 + \cdots + (2n-1)
\end{equation*}
\end{problem}

\begin{problem}
Prove that the sum of two even integers is even.
\end{problem}
\begin{proof}
Suppose that $a$ and $b$ are even integers.  This means that there are integers $x$ and $y$ such that $a = 2x$ and $b = 2y$.  Then $a + b = 2x + 2y = 2(x + y)$, by the distributive law.  Since $x$ and $y$ are integers, so is $x + y$, so by the definition of divisibility, $a + b$ is divisible by $2$.  But to be even means precisely to be divisible by $2$, so we conclude that $a + b$ is even.

As $a$ and $b$ were allowed to be arbitrary even integers in the paragraph above, this shows that the sum of two even integers is even.
\end{proof}

\begin{problem}
Prove that the sum of two odd integers is even.
\end{problem}

\begin{problem}
Use sum and product notation to rewrite the following expression without ellipses:
\begin{equation*}
1 + (1 \times 3) + (1 \times 3 \times 5) + (1 \times 3 \times 5 \times 7) + \cdots + (1 \times 3 \times 5 \times 7 \times \cdots \times (2n+1))
\end{equation*}
\end{problem}

\begin{problem}
Write a symbolic sentence meaning ``$x$ is not divisible by any perfect square'', using only quantifiers ($\forall$, $\exists$), variables, set membership ($\in$), logical connectives ($\wedge$, $\vee$, $\neg$, $\Rightarrow$, $\Leftrightarrow$), the set of integers ($\mathbb{Z}$), arithmetic (addition, subtraction, multiplication, division, exponentiation), and equality ($=$).
\end{problem}

\begin{problem}
Suppose that $x_1, x_2, \ldots$ is a sequence of real numbers.  We say that the sequence \emph{converges} to a real number $x$ if, for every positive real number $\epsilon$, there is positive integer $N$ such that, whenever $n$ is a positive integer greater than $N$, the distance between $x_n$ and $x$ is less than $\epsilon$.  

Rewrite this sentence symbolically, using only quantifiers ($\forall$, $\exists$), variables, set membership ($\in$), logical connectives ($\wedge$, $\vee$, $\neg$, $\Rightarrow$, $\Leftrightarrow$), the set of real numbers ($\mathbb{R}$), arithmetic (addition, subtraction, multiplication, division, exponentiation), absolute value, and equality and inequalities ($=, \neq, <, \leq, >, \geq$).
\end{problem}


\end{document}
