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\title{General Mathematics and Computational Science II}
\author{Exercise 17}
\date{April 24, 2007}
\maketitle

\begin{enumerate}

\item Show that for every sequence $a_n$ of nonnegative real numbers
there exists some $c>0$ such that
\[
  \sum_{n=1}^\infty a_n^2 \leq c \, \sum_{n=1}^\infty a_n \,.
\]
Can you give an estimate of the dependence of $c$ on $a_n$?

\item Let 
\[
  v(x) = \sum_{k=-\infty}^\infty \hat v_k \, \e^{\i k x} \,.
\]
Show that
\[
  \hat v_k = \frac1{2\pi} \int_0^{2\pi} \e^{-\i kx} \, v(x) \, \d x \,.
\]

\end{enumerate}
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