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\title{Math 3110: Homework 12 (the last!)}
%\author{Zach Norwood}
\date{\textcolor{navy}{Due Wednesday, 8 May 2019}}

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\begin{document}
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\noindent
You will be graded on both the accuracy and the clarity of your solutions.
One purpose of the homework is to give you an opportunity to practice your proof-writing skills.

You are welcome\,---\,encouraged, even!\,---\,to collaborate on homework, but you should not 
copy solutions from any source, nor should you submit anything that you don't understand.

\begin {problem}		
 Recall that $e$ is defined to be the number to which the series $\sum_{n\ge0} \frac{1}{n!}$ converges.
 \begin {enumerate} [(a)]
  \item
  Suppose that $b$ is a positive integer. Show that for large enough $n$, the quantity
  \[
   b \left( \frac{1}{n+1} + \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} + \cdots \right)
  \]
  lies strictly between $\frac{b}{n+1}$ and $\frac{b}{n}$.
  \item
  Supposing still that $b$ is a positive integer, consider the equation $e = \frac{a}{b}$.
  By multiplying each side by $n!b$ or otherwise, prove that $a$ is not an integer.
  Deduce that $e$ is irrational.
 \end {enumerate} 
\end {problem}

\begin {problem}
 Recall that the Fibonacci sequence $1,1,2,3,5,8,\dots$ is defined by $a_1 = a_2 = 1$ and $a_n = a_{n-1} + a_{n-2}$.
 \begin {enumerate} [(a)]
  \item
  Show that for every $n\in\N$ the inequality $\frac{a_{n+1}}{a_n} \le 2$ holds.
  \item
  Show that the power series
  \[
   \sum_{n\ge1} a_n x^{n-1} = 1 + x + 2x^2 + 3x^3 + 5x^4 + \cdots
  \]
  converges for $\abs{x} < 1/2$.
  \item
 Prove that for $\abs{x} < 1/2$ the power series in the previous part converges to
 \[
  f(x) = \frac{-1}{x^2 + x - 1}.
 \]
 (\emph{Hint:} Solve for $f(x)$ and compare coefficients.)
 \item
 Use partial fractions to obtain another power series for $f(x)$.
 \item
 The two power series must be the same. Conclude that
 \[
  a_n = \frac{1}{\sqrt5} \left( \frac{1 + \sqrt5}{2} \right)^n - \frac{1}{\sqrt5} \left( \frac{1 - \sqrt5}{2} \right)^n.
 \] 
 \item
 Reflect on the strange beauty of the formula from the previous part.
 \end {enumerate}
\end {problem}

\begin {problem}
 Which is greater, $e^\pi$ or $\pi^e$?
 
 (I'm not interested in what your calculator says. Give a \emph{proof} that your answer is correct,
 perhaps by considering the function $f(x) = \frac{ \ln x }{ x }$.)
\end {problem}

\begin {problem}
 Recall that we defined the natural logarithm function $\ln$ as the inverse of the exponential
 function $x\mapsto e^x$.
 \begin {enumerate} [(a)]
 \addtolength{\itemsep}{4pt}
  \item
  Prove that $\lim\limits_{h\to0^+} \frac{ \ln(1 + h) }{h} = 1$.
  (You cannot use l'H\^opital's rule here, since we did not prove it. But you don't need it.)
  
  \item
  Prove that $\lim\limits_{x\to\infty} x\ln(1 + 1/x) = 1$.
  
  \item
  Prove that $\lim\limits_{x\to\infty} (1 + 1/x)^x = e$.
  
  \item 
  Prove that for every $a\in\R$, $\lim\limits_{x\to\infty} (1 + a/x)^x = e^a$.
  (And reflect on the significance of this fact for compound interest.)
 \end {enumerate}
 
 (\emph{Hint}: Each part but the first can be deduced from the previous part.)
\end{problem}

\begin {problem}
 Recall that we defined $\pi$ to be the smallest positive solution to $\sin(x) = 0$.
 Prove that
 \[
  \pi = 2\int_{-1}^1 \sqrt{1 - x^2}\,dx,
 \]
 confirming that this $\pi$ is the area of the unit circle.
 
 (\emph{Hint}: You know what to do from calculus, but there are several things to prove.
 Most importantly, you must prove that your antiderivative of $\sqrt{1 - x^2}$ 
 has the properties you expect so that you can use the Fundamental Theorem of Calculus
 to evaluate the integral.)
\end {problem}

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