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\title{Advanced Calculus and \\Methods of Mathematical Physics}
\author{Homework 1}
\date{Due in class Tuesday, February 11, 2020}
\maketitle

\emph{Note:} Assignments marked (*) are \emph{not} for bonus credit.
They 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 Recall the mean value theorem of integral calculus: Suppose
$g \colon [a,b] \to \R$ is Riemann integrable and non-negative, and
$f \colon [a,b] \to \R$ is continuous.  Then there exists
$\xi \in [a,b]$ such that
\[
  f(\xi) \int_a^b g(x) \, \d x
  = \int_a^b f(x) \, g(x) \, \d x \,.
\]
Give an example each to show that the following assumptions cannot be
generally dropped:
\begin{enumerate}
\item $g$ is non-negative,
\item $f$ is continuous.
\end{enumerate}
\hfill(3)

\item *Recall Taylor's theorem in the following form.  Suppose $f \in
C^{n+1}(I)$ for some open interval $I$.  Then for all $a,x \in I$,
\[
  f(x) = \sum_{j = 0}^n \frac{f^{(j)}}{j!} \, (x-a)^j
         + R_n(x)
\]
where
\[
  R_n(x) = \frac1{n!} \int_a^x (x-t)^n \, f^{(n+1)}(t) \, \d t \,.
\]
\begin{enumerate}
\item Turn the derivation shown in class into a formal proof by
induction.
\item Show that the remainder can also be written as
\[
  R_n(x) = \frac{(x-a)^{n+1}}{n!}
  \int_0^1 (1-s)^n \, f^{(n+1)}(a + s(x-a)) \, \d s \,.
\]
\item Show that there exists $\xi \in [a,x]$ such that
\[
  R_n(x) = \frac{(x-a)^{n+1}}{(n+1)!} \, f^{(n+1)} (\xi) \,.
\]
(This expression is known as the \emph{Lagrange form} of the
remainder.) 
\end{enumerate}

\item *Show that 
\[ f_n (x) = \frac{\sin nx}n \]
is differentiable for every $n \in \mathbb N$.  Then show that this
sequence tends to $0$ pointwise, but the pointwise limit of the $f_n'$
does not exist for some $x \in \R$.

\item Find the Maclaurin series (i.e., the Taylor series centered at
$a=0$) of the function
\[
  f(x) = \e^{-x^2} \,.
\]
What it the interval of convergence?  Does it converge to the function
$f$ and why?
\hfill(4)

\item Find the limit of the series
\[
  \sum_{n=1}^\infty n^3 \, x^n \,.
\]
What is the radius of convergence?
\hfill(3)

\item *Consider a sequence of continuous functions $f_n\colon [a,b] \to [c,d]$
that converges uniformly to a function $f \colon [a,b] \to \R$.  Let
$\phi$ be continuous on $[c,d]$.  Show that the sequence $g_n = \phi
\circ f_n$ converges uniformly to $g = \phi \circ f$.

\end{enumerate}

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