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\title{Analysis II}
\author{Homework 5}
\date{Due in class Monday, March 11, 2019}
\maketitle

\begin{enumerate}


\item Let
\[
  \delta_n(x) =
  \begin{cases}
    0 & \text{for } \lvert x \rvert > 1/n \,, \\
    n/2 & \text{for } \lvert x \rvert \leq 1/n \,.
  \end{cases}
\]
Show that $\delta_n$ is a $\delta$-sequence, i.e., that
\[
  \int_{-\infty}^\infty \delta_n(x) \, \d x = 1
\]
for every $n \in \mathbb N$, and
\[
  \lim_{n \to \infty} \int_{-\infty}^\infty f(x) \, \delta_n(x) \, \d x
  = f(0)
\]
for every continuous function $f$ defined on $\R$.

\item Let $\{f_n\}$ be a sequence of twice differentiable functions on
$[0,1]$ such that $f_n(0)=f_n'(0)=0$ for all $n \in \mathbb N$ and
such that $\lvert f_n''(x) \rvert \leq 1$ for all $x \in [0,1]$ and $n
\in \mathbb N$.

Show that there exists a subsequence of $\{f_n\}$ which converges
uniformly on $[0,1]$.

\item (Rudin, Exercise 7.20.)  If $f$ is continuous on $[0,1]$ and if
\[
  \int_0^1 f(x) \, x^n \, \d x = 0
\]
for every $n = 0,1,\dots$, prove that $f(x) = 0$ on $[0,1]$.

\emph{Hint:} The integral of the product of $f$ with any polynomial is
zero.  Use the Weierstra\ss\ theorem to show that
\[
  \int_0^1 f^2(x) \, \d x = 0 \,.
\]

\item Find the radius of convergence for
\[
  \sum_{n=1}^\infty \frac{2^n}{\sqrt n} \, x^n \,.
\]


\item Find an explicit expression for the power series
\begin{enumerate}
\item $\displaystyle \sum_{n=1}^\infty n^3 \, x^n \,,$
\item $\displaystyle \sum_{n=1}^\infty \frac{x^n}n \,.$
\end{enumerate}
State the radius of convergence for each power series.

\item Find a power series representation centered at $2$ for
\[
  \frac1{4\,x - x^2 - 3} \,.
\]
What is the radius of convergence?
\end{enumerate}

\end{document}
