| 475 Spring 09 Homework
Math 475 Homework 9\\
\textsc{Due April 29th, 2009}
- 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.
- 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.)
- Write down an infinite series expression for $e$.
- 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.)
- Prove $\log_2 5$ is irrational. (Hint: what if it were equal to $p/q$� )
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| Math 475.01 Spring 09 [[475 Spring 09 Homework]] [[475 Spring 09 Big Problems]] Official Title. Math 475: Capstone Course . . . 3K - last updated 2009-08-20 21:44 by Eric Hsu |