## Geometric Series

Let $a,r\in\mathbb{R}$ be constants. Let $\{a_n\}$ be a sequence of real numbers with the following recursive definition:

$a_n=\begin{cases}a & \text{ if }n=1\\ r\cdot a_{n-1}&\text{ if }n>1\end{cases}$.

Therefore the sequence is given by:

$a,ar,ar^2,ar^3,ar^4,\dots$

Such a sequence is called a geometric sequence with common ratio $r$.

When we add up the terms a sequence we have a geometric sum:

$S_n=a+ar+ar^2+ar^3+\cdots ar^{n-1}$.

Here $S_n$ is the sum of the first $n$ terms.

We can find a formula for $S_n$ using the following ‘trick’:

$r\cdot S_n=ar+ar^2+ar^3+\cdots ar^n$

$\Rightarrow a+r\cdot S_n-ar^n=S_n$

$\Rightarrow S_n(r-1)=a(r^n-1)$

$\displaystyle \Rightarrow S_n=\frac{a(r^n-1)}{r-1}$.

### Exercises

Assuming that $|r|<1$, find a formula for the geometric series

$\displaystyle S_{\infty}=\lim_{n\rightarrow \infty}S_n$.

## Binary Numbers

### Exercises

• Write the following as fractions:

$0.1_2,\,0.11_2,\,0.101_2$.

• Use infinite geometric series to show that:
• $0.111\dots_2=1$
• $0.0111\dots_2=\frac12$
• $0.101010\dots_2=\frac23$

## Doubling Mapping

The doubling mapping $D:[0,1)\rightarrow [0,1)$ is given by:

$\displaystyle D(x)=\begin{cases}2x & \text{ if }x<1/2 \\ 2x-1 & \text{ of }x\geq 1/2\end{cases}$.

### Exercises

• Find the first six iterates of the point $x_0=\frac17$ under $D$.
• Find the first four iterates of the point

$x_0=\frac{1}{2}+\frac{1}{2^2}+\frac{0}{2^3}+\frac{1}{2^4}=0.1101_2$.

• Where $x$ has the binary representation

$x = 0.a_1a_2a_3a_4a_5a_6a_7a_8\dots$ ,

write down expressions for $D(x)$ and $D^5(x)$.

• Hence find points $y, z \in [0, 1]$ such that $y$ and $z$ agree to 5 binary digits but $D^N(y)$ and $D^N(z)$ differ in the first binary digit for some $N \in \mathbb{N}$.
• Describe the period-5 points of $D$.
• Let $w \in [0, 1]$ have a binary representation beginning $w = 0.01001\dots$  . Find a period-5 point $\gamma$ of $D$ such that $w$ and $\gamma$ agree to five binary digits.
• Find a $\delta \in [0, 1]$ such that there are iterates of $\delta$, $D^{n_1}(\delta),D^{n_2}(\delta),D^{n_3}(\delta)$, with $n_1, n_2, n_3 \in \mathbb{N}$, that agree with 0.111 , 0.101, and 0.010, to three binary
digits.

## Sensitivity to Initial Conditions

### Exercise

Let $f(x)=4x\cdot (1-x)$. Where $[0,1]$ is the set of states, and $f:[0,1]\rightarrow [0,1]$ the iterator function, by looking at the first seven iterates of $x_0=0.8$ and $y_0=0.81$, show that this dynamical system displays sensitivity to initial conditions [HINT:4*ANS*(1-ANS)]