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\title{Advanced Calculus and \\Methods of Mathematical Physics}
\author{Homework 2}
\date{Due in class Tuesday, February 18, 2020}
\maketitle

\emph{Note:} Assignments marked (*) will not be graded.  Do not turn
them in.  However, they will be discussed in the tutorial and example
solutions will be made available.

\begin{enumerate}

\item *Show that, for $\lvert x \rvert < 1$,
\[
  \arctan x = \sum_{n=0}^\infty (-1)^n \, \frac{x^{2n+1}}{2n+1} \,.
\]
\emph{Hint:} Represent $\arctan x$ by an integral.

\item Find a power series expansion centered at $0$ for
\[
  \frac1{(1+x^2)^2}
\]
and determine the radius of convergence.
\hfill(2)

\item Test the following series for uniform convergence on $\R$:
\begin{enumerate}
\item $\displaystyle \sum_{n=1}^\infty \frac{x}{(n+x^2)^2}$
\item $\displaystyle \sum_{n=1}^\infty \frac{\sin(nx)}n$
\end{enumerate}
\hfill (3)

\item (Rudin, Exercise 7.9.) Let $\{f_n\}$ be a sequence of continuous
functions which converges uniformly to a function $f$ on a set $E$.
Prove that
\[
  \lim_{n \to \infty} f_n(x_n) = f(x)
\]
for every sequence of points $x_n \in E$ such that $x_n \to x$, and $x
\in E$.
\hfill(2)

\item As in the lecture on February 14, let $\alpha \in R$ and define
the function
\[
  f(x) = \sum_{n=0}^\infty \binom{\alpha}{n} \, x^n
\]
on the interval $(-1,1)$.
\begin{enumerate}
\item Show that
\[
  f'(x) = \frac{\alpha}{1+x} \, f(x) \,.
\]
Justify the steps of your argument.

\item Show that
\[
  g(x) = \frac{f(x)}{(1+x)^\alpha}
\]
equals the constant $1$.

\item Conclude that $f(x) = (1+x)^\alpha$ on $(-1,1)$.

\end{enumerate}
\hfill(3)


\item *Suppose the series of complex numbers
\[
  \sum_{n=0}^\infty c_n
\]
is convergent.  Show that the power series
\[
  \sum_{n=0}^\infty c_n \, x^n 
\]
converges uniformly (but not necessarily absolutely!) for $x \in
[0,1]$.

\emph{Hints:} Consider the ``remainder'' series
\[
  r_n = \sum_{k=n}^\infty c_n
\]
and express the $c_n$ in terms of $r_n$.  Using this relation, estimate
\[
  \sum_{n=\ell}^m c_n \, x^n \,.
\]
You will encounter a geometric series as you go along, which provides
the uniformity on the interval $x \in [0,1]$.






\end{enumerate}

\end{document}
