\documentclass{article}

\title{\textbf{Homework 1}}
%\author{Enrique Trevi\~no}

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\begin{document}

\maketitle
\normalsize
\section{Chapter 1}
%\begin{problem}(\textbf{Exercise 1.1.1})\\
%Find the Fourier expansions for $f_1(x) = x$ and $f_2(x) = x^2$ over $[-\pi,\pi]$.
%\end{problem}
%\begin{sol}
%\end{sol}

\begin{problem}(\textbf{Exercise 1.1.3})\\
Using the Fourier series expansion for $x^2$ evaluated at $x = \pi$, show that
\[ \sum_{i=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.\]
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 1.1.18})\\
Show that
\[ 
\int_0^1 \!\left(\lim_{n\rightarrow \infty} n x e^{-n x^2}\right)\mathrm{d}x \ne \lim_{n\rightarrow\infty}\int_0^1 \left(n x e^{-n x^2}\right)\mathrm{d}x.
\]
\end{problem}
\begin{sol}
\end{sol}

\section{Chapter 2}
\begin{problem}(\textbf{Exercise 2.1.1})\\
Explain why if $P$ and $Q$ are partitions of the same interval and $Q$ is a refinement of $P, Q \supseteq P$, and if $f$ is any bounded function on this interval, then 
\[
\underline{S}(P; f) \le \underline{S}(Q; f)\le \overline{S}(Q; f)\le \overline{S}(P; f).
\]
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.1.2})\\
Consider the function $f$ defined by
\[
f(x) = \begin{cases} 
1, & x = 0,\\
x, & 0 < x < 1,\\
0, & x = 1.
\end{cases}
\]
Let $P$ be the partition $(0, 1/4, 1/2, 3/4, 1)$. Find the upper and lower Darboux sums, $\overline{S}(P; f)$ and $\underline{S}(P; f)$.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.1.3})\\
Using the function $f$ defined in Exercise 2.1.2 and given $\varepsilon = 1/2$, find a response $\delta$ so that for any partition $P$ into intervals of length less than $\delta$, the difference between $\overline{S}(P; f)$ and $\underline{S}(P; f)$ will be less than $1/2$.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.1.4})\\
Using the function $f$ defined in Exercise 2.1.2 over the interval $1/2\le x\le 1$, explain why no Riemann sum can equal the upper Darboux sum no matter what partition we choose.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.1.9})\\
Using the fact that a continuous function on a closed and bounded interval is uniformly continuous on that interval, prove that if $f$ is continuous on $[a. b]$, then $f$ is Riemann integrable over $[a,b]$.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.1.10})\\
Find the upper and lower Darboux integrals of the characteristic function of the rationals (Example 1.1 on page 3) over the interval $[0,1]$.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.1.11})\\
Prove that if $f$ is bounded on $[a, b]$, then 
\[
\underline{\int}_a^b f(x) \mathrm{d}x \le \overline{\int}_a^b f(x) \mathrm{d}x.
\]
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 2.1.15})\\
Define the function $n$ by 
\[
n(x) = \begin{cases}
1, &x = 1/n, n\in\mathbb{N}\\
0, &\mbox{otherwise}.
\end{cases}
\]
Show that $n$ is integrable over $[0,1]$ and that $\int_0^1 n(x)\mathrm{d}x = 0$.
\end{problem}
\begin{sol}
\end{sol}




\end{document} 