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{\bf \Large Math 112 Homework 9}\medskip

Due Thursday April 30, 2019, on Canvas.
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You are encouraged to discuss the homework problems with 
other students. However, what you hand in should reflect your 
own understanding of the material. You are NOT allowed to copy solutions 
from other students or other sources. {\it No late homeworks will be 
accepted.}

Please pay attention to the clarity and precision of your answers. Your
solutions to the problems should always consist of carefully written 
mathematical arguments. 

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{\bf Material covered:} Rudin pages 149--154, 159--162, 178--181.
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{\bf Problem 1.} (5 points) Rudin Chapter 7 Problem 20.
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{\bf Problem 2.} (5 points) Rudin Chapter 7 Problem 23.
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{\bf Problem 3.} (5 points) Rudin Chapter 8 Problem 10.

Follow the hint page 198; you will need to justify the various statements
made in the hint. For the first inequality, a careful explanation is probably
better than a lengthy formal argument. For the last inequality, consider
the derivative of $(1-x)e^{2x}$. Once you are done with the hint, don't
forget to prove the result itself!
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{\bf Problem 4.} (5 points) Rudin Chapter 8 Problem 20.

Follow the given outline of argument, proving the various statements therein. 
In order to show that $f(x)\le \log x\le g(x)$, argue over each subinterval 
by using differentiation. Remember that $\int \log x\,dx$ can be computed
by integration by parts.

Note: Stirling's formula asserts that $n!\sim n^n e^{-n} \sqrt{2\pi n}$,
i.e.\ $\displaystyle \lim_{n\to\infty} \frac{n!}{n^n e^{-n}\sqrt{2\pi n}}=1$.

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