Week 5 Exercises, due 26 October 2021

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Exercises 5.7 Week 5 Exercises, due 26 October 2021

Let \((a_n)_{n=0}^{\infty}\) be a sequence of positive real numbers. For every \(n\in\NN_0\text{,}\) we may define the expression

\begin{equation*} [a_0, a_1, \ldots, a_n] = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{\ddots +\cfrac{1}{a_n}}}}\text{.} \end{equation*}

More systematically, we define this expression inductively: for \(n=0\text{,}\) we write \([a_0]=a_0\text{;}\) now for every \(n \in \NN_0\text{,}\) we define

\begin{equation*} [a_0, a_1, \dots, a_n]=\left[a_0,a_1,\dots,a_{n-2},a_{n-1}+\frac{1}{a_n}\right]\text{.} \end{equation*}

1.

Define two auxiliary sequences \((u_n)_{n=-2}^{\infty}\) and \((v_n)_{n=-2}^{\infty}\) of real numbers recursively as follows. Let

\begin{equation*} u_{-2} = 0 \text{ and } u_{-1} = 1 \end{equation*}

and

\begin{equation*} v_{-2} = 1 \text{ and } v_{-1} = 0\text{.} \end{equation*}

For any natural number \(n\in \NN_0\text{,}\) let

\begin{equation*} u_n = a_nu_{n-1}+u_{n-2} \text{ and } v_n = a_nv_{n-1}+v_{n-2}\text{.} \end{equation*}

Prove that

\begin{equation*} [a_0,a_1,\dots,a_n] = \frac{u_n}{v_n}\text{.} \end{equation*}

2.

If \(a_n = 1\) for every natural number \(n\text{,}\) then what are the numbers \(u_n\) and \(v_n\text{?}\) If you don't know a name for these sequences, that's ok; please just compute the first ten terms of each sequence.

3.

Prove that for any \(n \in \NN_0\text{,}\) we have

\begin{equation*} u_nv_{n-1} - u_{n-1}v_n = (-1)^{n-1}\text{.} \end{equation*}

4.

For ease of notation, let us write \(x_n = [a_0,a_1, \dots,a_n]\) for \(n\in\NN_0\text{.}\) Prove that the subsequences

\begin{equation*} (x_{2n})_{n=0}^{\infty} \text{ and } (x_{2n+1})_{n=0}^{\infty} \end{equation*}

are each bounded and monotonic, hence convergent. Write \(A = \lim_{n\to \infty} x_{2n}\) and \(B = \lim_{n\to \infty} x_{2n+1}\text{.}\)

5.

Assume now that for every \(n\in\NN_0\text{,}\) one has \(a_n \in \NN\text{.}\) Prove that \(A=B\text{.}\) Conclude that the sequence \((x_n)_{n=0}^{\infty}\) converges. Write \([a_0,a_1,\dots]\) for the limit.

6.

Let \(s,t \in \NN\text{.}\) Find an expression for \([s,t,s,t,\dots]\text{.}\) (Hint: you may use the quadratic formula.)