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diff --git a/dviware/catdvi/test/texmext.tex b/dviware/catdvi/test/texmext.tex new file mode 100644 index 0000000000..35f6d1ceca --- /dev/null +++ b/dviware/catdvi/test/texmext.tex @@ -0,0 +1,40 @@ +\documentclass{article} + +\begin{document} + +\section*{Some problems for the DVI class} + +\begin{enumerate} +\item +Let $f$ be a real-valued, nonnegative function on the positive half-axis. +For every positive integer $i$, put $a_i := f(i)$. +Assume that $f$ is monotonically decreasing. Then +$\sum_{i=1}^{\infty} a_i$ is finite iff $\int_1^{\infty} f(x)\,dx$ is. + +\item +And here's a harder one: consider nonnegative solutions $\epsilon$, +$\delta$ of the equation +\[ +\bigg(1 + \epsilon \Big(1 + \epsilon \big(1 + \epsilon (1 + \epsilon + ) \big) \Big) \bigg) = +\sqrt{1 + \delta \sqrt {1 + \delta \sqrt {1 + \delta \sqrt{1 + \delta}}}} +\] +Is it true (and if so, in which sense) that $\epsilon \ll \delta$ as +$\delta \rightarrow 0$ ? What happens if the equation is changed to +\[ +\bigg(1 + \epsilon \Big(1 + \epsilon \big(1 + \epsilon (1 + \epsilon + )^2 \big)^2 \Big)^2 \bigg)^2 = +\sqrt[4]{1 + \delta \sqrt[4] {1 + \delta \sqrt[4] {1 + \delta \sqrt[4]{1 + \delta}}}} +\] + +\item +Prove that the series +\[ +\sum_{n=1}^{\infty} \frac{(-1)^n\,n}{(n+1)^2} +\] +does converge. (Hint: don't try to use problem 1). +\end{enumerate} +Solutions have to be submitted by yesterday, 0730~AM at my office in +ascii, latin1 and utf-8 encoding. + +\end{document} |