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\title{Math 412  Homework 11}
\date{Due date: Nov 13, 2015}
\author{your name}


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Solve the following problems. Please remember to use complete sentences
and good grammar. 



\begin{enumerate}

\item Show that if $a_0>0$, then $p_k/p_{k-1}=[a_k, a_{k-1}, \ldots, a_1, a_0]$ and $q_k/q_{k-1}=[a_k, a_{k-1}, \ldots, a_2, a_1]$, where $C_{k-1}=p_{k-1}/q_{k-1}$ and $C_k=p_k/q_k$, $k\ge 1$, are successive convergence of the continued fraction $[a_0, a_1, \ldots, a_n]$. 
   
\item Let $\alpha>1$ be an irrational number. Show that the $k$-th convergent of the simple continued fraction of $1/\alpha$ is the reciprocal of the $(k-1)$-th convergent of the simple continued fraction of $\alpha$. 


\item Let $\alpha$ be an irrational number and let $p_j/q_j$ be the $j$-th convergent of the simple continued fraction expansion of $\alpha$. Show that at least one of any three consecutive convergence satisfies the inequality $|\alpha-p_j/q_j|<1/(\sqrt{5}q_j^2)$.  Conclude that there are infinitely many rational numbers $p/q$, where $p$ and $q$ are integers with $q\not=0$, such that $|\alpha-p/q|<1/(\sqrt{5}q^2)$.  

\item Show that the simple continued fraction $\sqrt{d}$, where $d$ is a positive integer, has period length one if and only if $d=a^2+1$, where $a$ is a nonnegative integer.  

\item Show that if $p_k/q_k$ is a convergent of the simple continued fraction expansion of $\sqrt{d}$, then $|p_k^2-dq_k^2|<1+2\sqrt{d}$. 

\item (Bonus) Let $k$ be a positive integer and $D_k=(3^k+1)^2+3$. Show that the simple continued fraction of $\sqrt{D_k}$ has a period of length $6k$. 



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