\documentclass[addpoints, 11pt]{exam}
\setlength{\headsep}{0.125in}
\setlength{\unitlength}{1in}
%
\pagestyle{head}
%
%
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
%
\newcommand{\myCourseNumber}{Math 142B}
\newcommand{\myAssignmentDueDate}{11:00pm Thursday, March 14, 2023}
\newcommand{\myAssignmentTitle}{Homework Assignment 5}
%
%
\begin{document}
%
\firstpageheader{\bfseries \myCourseNumber \enspace \myAssignmentTitle \\ Due \myAssignmentDueDate }{}{}
%
\runningheader{{\bfseries (page \textit{\thepage}\ of \textit{\numpages})}}{}{}
%
%\firstpagefooter{}{}{(This exam is worth {\numpoints} points.)}
%

\begin{questions}

%%Ross 23.4
\question For \ $n = 0, 1, 2, 3, \ldots$ \ let \ $\displaystyle a_n = \left[ \frac{4 + 2(-1)^n}{5} \right]^n$.
\begin{parts}
\part Find
\begin{subparts}
\subpart $\displaystyle \limsup (a_n)^{\frac{1}{n}}$
\subpart $\displaystyle \liminf (a_n)^{\frac{1}{n}}$
\subpart $\displaystyle \limsup \left| \frac{a_{n+1}}{a_n} \right|$
\subpart $\displaystyle \liminf \left| \frac{a_{n+1}}{a_n} \right|$
\end{subparts}
\part Do the series $\sum a_n$ and $\sum (-1)^n a_n$ converge? Briefly justify your answers.
\part Find the radius of convergence and exact interval of convergence of the power series $\sum a_n x^n$ with $a_n$ as above.

\end{parts}

%%Ross 23.5
\question Let $\sum a_n x^n$ be a power series with radius of convergence $R$. Prove:
\begin{parts}
\part If all the coefficients $a_n$ are integers and $a_n \not= 0$ for infinitely many $n$, then $R \leq 1$.
\part If \ $\limsup |a_n| > 0$, \ then $R \leq 1$.
\end{parts}

%%Ross 23.6
\question \begin{parts}
\part Suppose $\sum a_n x^n$ has finite radius of convergence $R$ and $a_n \geq 0$ for all $n$. Show that if the series converges at $x = R$, then it also converges at $x = -R$.
\part Exhibit an example of a power series whose interval of convergence is exactly $(-1, 1]$. \newline  (\textbf{Note:} ``Exhibit'' means ``Show that the example has the required properties.'')
\end{parts}

%%Ross 26.4
\question \begin{parts}
\part Verify that \ $\displaystyle e^{-x^2} = \sum_{n=0}^\infty \frac{(-1)^n}{n!} x^{2n}$ \ for all $x \in \mathbb{R}$, since \ $\displaystyle e^x  = \sum_{n=0}^\infty \frac{1}{n!} x^n$ \ for all $x \in \mathbb{R}$.
\part Write \ $\displaystyle F(x) = \int_0^x e^{-t^2} \, dt$ \ as a power series.  Be sure to briefly explain how you know that the power series for $F(x)$ converges for all $x \in \mathbb{R}$.
\end{parts}

%%Ross 26.5
\question Let \ $\displaystyle f(x) = \sum_{n=0}^\infty \frac{1}{n!} x^n$ \ for $x \in \mathbb{R}$.  Using only the properties of power series, show that $f' = f$.

%%Ross 26.6
\question For $x \in \mathbb{R}$, let
\begin{align*}
s(x) &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} \\
c(x) &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}
\end{align*}
Prove:
\begin{parts}
\part $s' = c$ \ and \ $c' = -s$.
\part $\displaystyle \left( s^2 + c^2 \right)' = 0$.
\part $\displaystyle s^2 + c^2 = 1$.
\end{parts}

\newpage

%%Ross 26.8
\question \begin{parts}
\part Show that \ $\displaystyle \sum_{n=0}^\infty (-1)^n x^{2n} = \frac{1}{1 + x^2}$ \ for $x \in (-1,1)$.
\part Show that \ $\displaystyle \arctan(x) = \sum_{n=0}^\infty \frac{(-1)^n}{2n + 1} x^{2n + 1}$ \ for $x \in (-1,1)$.
\part Show that the equality in (b) also holds for $x = 1$.  Use this to find a fun formula for $\pi$.
\part What happens at $x = -1$?
\end{parts}

%%Ross 36.6
\question Prove the following \emph{comparison tests}.  Let $f$ and $g$ be continuous functions on $(a,b)$ such that $0 \leq f(x) \leq g(x)$ for all $x \in (a,b)$ and where $a$ could be $-\infty$ and $b$ could be $+\infty$.
\begin{parts}
\part If \ $\displaystyle \int_a^b g(x) \, dx < \infty$, then $\displaystyle \int_a^b f(x) \, dx < \infty$.
\part If \ $\displaystyle \int_a^b f(x) \, dx = +\infty$, then $\displaystyle \int_a^b g(x) \, dx = +\infty$.
\end{parts}

%%Ross 36.7
\question \begin{parts}
\part Using a comparison test, show that $\displaystyle \int_{-\infty}^\infty e^{-x^2} \, dx < \infty$.
\part Show that $\displaystyle \int_{-\infty}^\infty e^{-x^2} \, dx = \sqrt{\pi}$.
\end{parts}

%%Ross 36.8
\question Suppose $f$ is continuous on $(a,b)$, where $a$ could be $-\infty$ and $b$ could be $+\infty$.  Show that if \ $\displaystyle \int_a^b |f(x)| \, dx < \infty$, then the integral \ $\displaystyle \int_a^b f(x) \, dx$ \ exists and is finite.

\end{questions}

\end{document}
