\documentclass[12pt]{article} \usepackage{amssymb,amsmath,graphicx,hyperref} \hypersetup{ pdfnewwindow=true, pdffitwindow=false } \pagestyle{empty} \begin{document} \begin{center} \textbf{Math 475 Homework 9}\\ \textsc{Due April 29th, 2009} \end{center} \begin{enumerate} \item We've defined $e = \lim_{n \to \infty} (1 + \frac{1}{n})^n$. There is another famous series expression for $e$, which we can use to prove $e$ is irrational. \begin{enumerate} \item Write down the Taylor Series for the function $e^x$. (You can look it up. I'll assume you derived it in Calculus 2 and can prove it converges everywhere.) \item Write down an infinite series expression for $e$. \item Assuming $e$ is genuinely equal to the infinite series, prove $e$ is irrational. (Hint: if $e = p/q$, multiply through by $(q!)$. This should lead to an integer part plus something you can prove is less than 1.) \end{enumerate} \item Prove $\log_2 5$ is irrational. (Hint: what if it were equal to $p/q$…? ) \end{enumerate} \end{document}