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\newcommand{\myCourseNumber}{Math 142B}
\newcommand{\myAssignmentDueDate}{11:00pm Thursday, May 23, 2024}
\newcommand{\myAssignmentTitle}{Homework Assignment 4}
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\begin{document}
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\firstpageheader{\bfseries \myCourseNumber \enspace \myAssignmentTitle \\ Due \myAssignmentDueDate }{}{}
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\runningheader{{\bfseries (page \textit{\thepage}\ of \textit{\numpages})}}{}{}
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%\firstpagefooter{}{}{(This exam is worth {\numpoints} points.)}
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\begin{questions}

%% Ross 24.9
\question Consider $f_n(x) = n x^n (1-x)$ for $x \in [0,1]$.
\begin{parts}
\part Find $\displaystyle f(x) = \lim_{n \to \infty} f_n(x)$.
\part Does $f_n \to f$ uniformly on $[0,1]$? Be sure to justify your answer.
\part Does $\displaystyle \int_0^1 f_n(x) \, dx$ converge to $\displaystyle \int_0^1 f(x) \, dx$?  Be sure to justify your answer.
\end{parts}

%% Ross 24.12
\question Prove that a sequence $(f_n)$ of functions on a set $S \subseteq \mathbb{R}$ converges uniformly to a function $f$ on~$S$ \ if and only if
\ $\displaystyle \lim_{n \to \infty} \sup \left\{ \left| f(x) - f_n(x) \right| \ \mid \ x \in S \right\} = 0$.

%% Ross 24.13
\question Prove that if $(f_n)$ is a sequence of functions uniformly continuous on an interval $(a,b)$ and if $f_n \to f$ uniformly on $(a,b)$, then $f$ is also uniformly continuous on $(a,b)$.

%% Ross 24.17
\question Let $(f_n)$ be a sequence of continuous functions on $[a,b]$ that converges uniformly to $f$ on $[a,b]$.  Show that if $(x_n)$ is a sequence in $[a,b]$ with $x_n \to x$, then $f_n(x_n) \to f(x)$.
~
%% Ross 25.4
\question Let $(f_n)$ be a sequence of functions on a set $S \subset \mathbb{R}$ such that $f_n \to f$ uniformly on $S$.  Prove that $(f_n)$ is uniformly Cauchy on $S$.

%% Ross 25.5
\question Let $(f_n)$ be a sequence of bounded functions on a set $S$ such that $f_n \to f$ uniformly on $S$. Prove that $f$ is bounded on $S$.

%% Ross 25.6
\question
\begin{parts}
\part Show that if $\sum |a_k| < \infty$, then $\sum a_k x^k$ converges uniformly on $[-1,1]$ to a continuous function.
\part Does $\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} x^n$ represent a continuous function on $[-1,1]$?
\end{parts}

%% Ross 25.9
\question
\begin{parts}
\part Let $0 < a < 1$.  Show that the series $\displaystyle \sum_{n=0}^\infty x^n$ converges uniformly on $[-a,a]$ to $\displaystyle \frac1{1-x}$.
\part Does the series $\displaystyle \sum_{n=0}^\infty x^n$ converge uniformly on $(-1,1)$ to $\displaystyle \frac1{1-x}$?
\end{parts}

%% Ross 25.15b
\question Let $(f_n)$ be a sequence of continuous functions on $[a,b]$ such that $\left(f_n\left(x\right)\right)$ is an increasing sequence of real numbers for each $x \in [a,b]$.  Prove that if $f_n \to f$ pointwise on $[a,b]$ and if $f$ is continuous on $[a,b]$, then $f_n \to f$ uniformly on $[a,b]$.  (This is called \emph{Dini's theorem}.)

%%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}

\end{questions}

\end{document}
