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\title{General Mathematics and Computational Science I}
\author{Exercise 15}
\date{November 1, 2007}
\maketitle

\begin{enumerate}

\item Solve Problem 17 from Ivanov, p.~53.

\item (From Ivanov, p.\ 53) Show that
\[
  \sqrt[n]{n} > \sqrt[n+1]{n+1}
\]
for $n \geq 3$.

\emph{Hints:}
\begin{enumerate}
\item Consider the expression $(1+n)^n \, n^{\tfrac{n+1}n}$, use the
  AM-GM inequality;
\item Notice that
\[
  3^{-\tfrac13} + \tfrac14 \approx 0.943361 < 1 \,;
\]
\item part of the argument is inductive.
\end{enumerate}
(Extra points if you find a proof which is more direct than the
one suggested by the hints above.)

\item Prove \emph{Jensen's inequality}
\[
  \biggl( \sum_{i=1}^n a_i^p \biggr)^{\tfrac1p}
  \geq
  \biggl( \sum_{i=1}^n a_i^q \biggr)^{\tfrac1q}
\]
whenever $0<p \leq q$ and $a_1, \dots, a_n \geq 0$.  If $p<q$, then
equality occurs if and only if at most one of the $a_i$ is nonzero.

\emph{Hint:}  A proof is sketched on p.\ 57 in Ivanov.  You should
fill in the gaps and write out a complete, self-contained proof.
\end{enumerate}
 
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